RETAINING  WALL! 


FOR     FAPTH 

PvJi\     LLntfxJ  I  I 


E 


LIBRARY 


UNIVERSITY  OF  CALIFORNIA. 


Class 


WORKS  OF  PROF.  M.  A.  HOWE 

PUBLISHED    BY 

JOHN  WILEY  &  SONS. 


The  Design  of  Simple   Roof-trusses  in  Wood  and 
Steel. 

With  an  Introduction  to  the  Elements  of  Graphic 
Statics.  8vo,  vi -fi29  pages,  67  figures,  and  3  folding 
plates.  Cloth,  $2.00. 

Retaining  walls  for  Earth. 

Including  the  Theory  of  Earth-pressure  as  Developed 
from  the  Ellipse  of  Stress-  With  a  Short  Treatise  on 
Foundations.  Illustrated  with  Examples  from  Prac- 
tice. Third  edition,  revised  and  enlarged.  izrno, 
cloth,  $1.25. 

A  Treatise  on  Arches. 

Designed  for  the  use  of  Engineers  and  Students  in 
Technical  Schools.  8vo,  xxv  +  351  pages,  74  figures. 
Cloth,  $4.00. 


RETAINING- WALLS 

FOR  EARTH. 


INCLUDING 

THE   THEORY  OF  EARTH-PRESSURE 

AS  DEVELOPED  FROM  THE 

ELLIPSE   OF  STRESS. 

WITH 

A  SHORT  TREATISE  ON  FOUNDATIONS,  ILLUSTRATED 
WITH  EXAMPLES  FROM  PRACTICE. 


BY 

MALVERD   A.   HOWE,  C.E., 

Professor  of  Civil  Engineering,  Pose  Polytt  clinic  Institute  f 
Member  American  Society  of  Civil  Engineers. 


THIRD  EDITION,  REVISED  AND  ENLARGED. 
FIRST   THOUSAND. 


NEW  YORK: 

WILEY  &  SONS. 
LONDON  :   CHAPMAN  &  HALL,  LIMITED. 
1905. 


Copyright,  1896, 

BY 

MALVERD  A.  HOWE. 


ROBERT  DRUMMOND,   ELKCTROTYPKR  AND  PRINTER,  NEW  YORK. 


CONTENTS. 


THEORY   OF  EARTH-PRESSURE. 

PAGE 

Preliminary  Principles 1 

Resultant  of  Principal  Stresses.  Case  1 2 

Case  II 3 

Caselll 3 

Earth-pressure  against  a  Vertical  Plane 9 

Direction  of  Resultant  Earth-pressure  against  a  Vertical  Plane. .  11 
Intensity  of  Earth-pressure  against  a  Vertical  Plane  at  any  given 

Depth 12 

Average  Intensity  of  Earth-pressure  against  a  Vertical  Plane. ...  13 
Graphical  Construction  for  finding  Thrust  of  Earth  against  any 

Plane 13-15 

General  Formula  for  the  Thrust  of  Earth 15-17 

"  "  "  "  Direction  of  the  Resultant 

Earth-pressure 18 

Plane  of  Rupture 18 

Reliability  of  Theory 19 

Earth  Sloping  Down  and  Away  from  Wall — Special  Method. . .  21 


FORMULAS  FOR  EARTH-PRESSURE. 

Recapitulation. 

General  Formula 23 

Surface  of  the  Earth  inclined,  a  —  0 24 

Surface  of  the  Earth  Parallel  to  the  Surface  of  Repose 24 

Surface  of  the  Earth  Horizontal 25 

ill 


\ 


IV  CONTENTS. 

PAGE 

Fluid  Pressure 25 

Graphical  Construction  for  determining  the  Thrust  of  Earth 
against  any  Plane 25 


STABILITY  OF  TRAPEZOIDAL  WALLS. 

Stability  against  Overturning 29 

"      Sliding 29 

"  "      Crushing  of  Material 80 

Determination  of  the  Width  of  Base  of  a  Trapezoidal  Wall 33 


FORMULAS  FOR  TRAPEZOIDAL  AND  TRIANGULAR 
WALLS. 

General  Formula  for  Trapezoidal  Walls 34 

Formula  for  Vertical  Wall  34 

"          "a  Wall  with  a  Vertical  Back  resisting  a  Normal 

Earth-pressure 35 

General  Formula  for  Tria'ngular  Walls 36 

Special  Cases  of  Triangular  Walls 36 


FOUNDATIONS  FOR  WALLS  RETAINING  EARTH. 

General  Discussion 37 

Depth  of  Foundations 38 

Depth  of  Foundation  when  the  Intensity  of  the  Pressure  upon 

the  Base  is  Uniform 39 

Depth  of  Foundation  when  the  Intensity  of  the  Pressure  upon 

the  Base  is  Uniformly  Varying 39-40 

Depth  of  Foundation  when  the  Earth  has  Different  Depths  on 

Opposite  Sides  of  the  Wall 41 

Determination  of  the  Breadth  of  the  Base  of  a  Trapezoidal 

Foundation 42 

Abutting  Power  of  Earth 43 

Bearing  Power  of  Earth 43 


CONTENTS.  T 
EXAMPLES. 

PAOE 

Examples  illustrating  the  Application  of  Formulas  for  Earth- 
pressure,  Depth  of  Foundations,  etc 44-60 

Examples  of  Retain  ing- wall  Profiles 61-65 

FOUNDATIONS. 

Foundations  upon  Rock 66 

Maximum  Deviation  of  Resultant  Pressure  from  the  Centre  of 

the  Base  of  the  Foundation 67 

Ultimate  Compressive  Strengths  of  Stone 68 

Foundations  upon  Earth 68 

Firm  Earth 68 

Determination  of  the  Breadth  of  a  Symmetrical  Trapezoidal 

Foundation 73 

Examples 74 

Unsymmetrical  Distribution  of  Pressure  upon  the  Base  of  the 

Foundation 75 

Formula  for  Breadth  of  the  Base 76 

Projection  of  Foundation  Courses,  etc 77,  78 

Table  of  Safe  Projections  of  Courses 79 

Foundations"  upon  Soft  Earth 78 

Projection  of  Steel  or  Iron  Beams  used  in  Foundations 80 

Table  of  Safe  Projections 81 

Pile  Foundations 81 

Formula  for  Minimum  Depth  of  Pile 83 

Trautwine's  Formula 85 

Engineering  News  Formula 85 

Screw  Piles 85 

Sheet  Piles 86 

FOUNDATIONS   UNDER  WATER  AND  DEEP  FOUN- 
DATIONS. 

Coffer-dams 87 

Timber  Cribs 87 

Open  Caissons 88 


VI  CONTENTS. 

PAGE 

Gushing  Cylinder  Piers 89 

Pneumatic  Caissons ;,. . .     89 

TYPES  OP  EXISTING  FOUNDATIONS 90-97 

REFERENCES,  LIST  OF 99-102 

DIAGRAM  I....  103 

TABLES. 

Weights  of  Materials 106 

Angles  and  Coefficients  of  Friction 107 

Values  of  Functions  B,  C,  D,  and  E 108-110 

Natural  Sines,  Cosines,  Tangents,  and  Cotangents 111-132 


PREFACE   TO   THE    SECOND  EDITION. 

THE  first  edition  of  this  work  was  based  upon  the 
theory  advanced  by  Prof.  Weyraucb  in  1878,  but  owing 
to  the  length  of  the  demonstrations  used  by  him,,  it 
was  thought  advisable  to  present  different  and  shorter 
demonstrations  in  this  edition.  To  show  that  the  new 
demonstrations  give  identical  results  with  those  obtained 
by  Prof.  Weyrauch,  his  demonstrations  have  been  given  in 
an  appendix  as  they  appeared  in  the  first  edition. 

The  new  demonstrations  are  based  upon  the  theory  first 
advanced  by  Prof.  Rankine  in  1858.  Those  readers  who 
are  familiar  with  Rankine's  Ellipse  of  Stress  can  omit 
pages  1  to  9,  inclusive,  in  following  the  demonstrations. 

An  attempt  has  been  made  to  present  the  theory  in  a 
shape  easily  followed  by  those  who  have  only  a  knowledge 
of  algebra,  geometry,  and  trigonometry;  whenever  cal- 
culus has  been  resorted  to,  the  work  has  been  simplified  as 
much  as  possible.  For  convenience  in  practice,  the  formu- 
las have  been  arranged  in  a  condensed  shape  in  Part  I, 
and  are  followed  by  numerous  examples  illustrating  their 
application. 

The  values  of  various  coefficients  have  been  computed 
and  tabulated  and  will  be  found  to  very  materially  decrease 
the  labor  of  substitution  in  the  formulas. 

vii 


vm  PREFACE. 

It  is  hoped  that  the  introduction  of  a  brief  treatment  of 
the  supporting  power  of  earth  in  the  case  of  foundations, 
as  well  as  the  formula  for  determining  the  breadth  of  the 
base  of  a  retaining-wall,  will  prove  acceptable. 

For  valuable  help  in  the  verification  of  proofs  of  formu- 
las, and  the  critical  reading  of  the  whole  text,  I  acknowl 
edge  the  kind  assistance  of  Prof.  Thos.  Gray. 

M.  A.  H. 


PREFACE  TO  THE  THIRD  EDITION. 


IN  this  edition  a  large  number  of  examples  illustrating 
several  profiles  of  retaining-walls  and  types  of  foundations 
selected  from  existing  structures  have  been  included.  The 
Appendix  of  the  second  edition  has  been  replaced  by  a 
treatise  on  Foundations  sufficiently  short  and,  the  author 
believes,  sufficiently  complete  for  the  use  of  technical 
schools,  if  judiciously  supplemented  by  lectures  or  refer- 
ences to  descriptions  of  existing  structures. 

M.  A.  H. 
TERRE  HAUTE,  IND.,  Nov.  1896. 


NOMENCLATIVE. 


0  =  the  angle  of  repose,  or  the  maximum  angle  which 
any  force  acting  upon  any  plane  within  the  mass 
of  earth  can  make  with  the  normal  to  the  plane. 

e  =  the  angle  made  by  the  surface  of  the  earth  with  the 
horizontal;  e  is  positive  when  measured  above  and 
negative  when  measured  below  the  horizontal. 

a  =  the  angle  which  the  back  of  the  wall  makes  with 
the  vertical  passing  through  the  heel  of  the  wall; 
a  is  positive  when  measured  on  the  left  and  nega- 
tive when  measured  on  the  right  of  the  vertical. 

6  =  the  angle  which  the  direction  of  the  resultant  earth- 
pressure  makes  with  the  horizontal. 

0'  =  the  angle  of  friction  between  the  wall  and  its  foun- 
dation. 

0"  =  the  angle  of  friction  between  the  back  of  the  wall 
and  the  earth. 

H  =  the  vertical  height  of  the  wall  in  feet. 

h  =  the  depth  of  earth  in  feet  which  is  equivalent  to  a 

given  load  placed  upon  the  surface  of  the  earth. 
B'  =  the  width  in  feet  of  the  top  of  the  wall. 

B  =  the  width  in  feet  of  the  base  of  the  wall. 

Q  —  the  distance  in  feet  from  the  toe  of  the  wall  to  the 
point  where  R  cuts  the  base. 

ix 


X  NOMENCLATURE. 

P  =  the  resultant  earth-pressure  in  pounds  against  a  ver- 
tical wall. 
E  =  the  resultant  earth-pressure  in  pounds  against  any 

wall. 
R  =  the  resultant  pressure  in  pounds  on  the  base  of  the 

wall. 

G  —  the  total  weight  in  pounds  of  material  in  the  wall. 
y  =  the  weight  in  pounds  of  a  cubic  foot  of  earth. 
W  =  the  weight  in  pounds  of  a  cubic  foot  of  wall. 
p  =  the  intensity  of  the  pressure  in  pounds  on  the  base 

of  the  wall  at  the  toe. 
p'  =  the  intensity  of  the  pressure  in  pounds  on  the  base 

of  the  wall  at  the  heel. 
po  =  the  average  intensity  of  the  pressure  in  pounds  on 

the  base  of  the  wall. 
x  =  H  tan  a. 
x"  and  xf  =  depth  of  the  base  of  the  foundation  below  the 

earth  surface. 
B"  =  breadth  of  the  base  of  the  foundation. 

o  =  the  oifset  of  a  foundation  course. 
G'  =  the  total  weight  of  the  material  above  the  base  of 

the  foundation. 


THEORY  OF  EARTH-PRESSURE. 


Preliminary  Principles. — Before  demonstrating  the  gen 
eral  formula  for  the  thrust  of  earth  against  a  wall,  it  will 
be  necessary  to  establish  the  relations  between  the  stresses 
in  an  unconfined  and  homogeneous  granular  mass. 

*  In  Fig.  1  let  ABC  be  any  small  prism  within  a  granu- 

H  F       Q         .E 


Fio.  1. 

lar  mass  which  is  in  equilibrium  un^er  the  action  of  the 
three  stresses  P,  Q,  and  R,  having  the  intensities^,  q,  and 
r  respectively. 

*  In  all  the  demonstrations  which  follow,  the  dimension  perpen 
dicular  to  the  page  will  be  considered  as  unity. 

1 


2  THEORY  OF  EARTH-PRESSURE. 

Let  0  represent  the  angle  of  inclination  of  the  plane  CB 
with  AB,  and  the  angle  at  A  be  a  right  angle. 

The  planes  AB  and  AC  are  called  planes  of  principal 
stress,  and  P  and  Q  are  called  principal  stresses. 

CASE  I.  If  tliQ  principal  stresses  are  of  the  same  kind 
and  their  intensities  the  same,  then  will  the  resultant  stress 
on  any  third  plane  he  normal  to  that  plane  and  its  inten- 
sity be  equal  to  that  of  either  principal  stress. 

In  Fig.  1,  for  convenience,  let  AB  =  1,  then  A  C  =  tan  6, 

and  CB  —  --  -„.    Hence 

P  =  p,  Q=:qtanO  =  p  tan  6,  since  p  =  q,  and  R  =  --  fi  . 

COS  u 

Since  P,  Q,  and  R  are  in  equilibrium,  they  will  form  a 
closed  triangle,  as  shown  on  the  right  in  Fig.  1.  Hence 


or 

tan8  e  =/>'(!  +  tan8 


Also,  Rco8FDE=P, 


or  —-7,  cos  FDE  =  p-,    but  r  =p. 

cos  u 


Hence  cos  6  =  cos  FDE  =  cos  HDG; 

.'.  HDG  =  6    and     R  is  normal  to  CB. 


THEORY  OF  EARTH-PRESSURE.  3 

CASE  II.  If  the  principal  stresses  are  not  of  the  same 
kind  but  their  intensities  the  same,  then  will  the  resultant 
make  the  angle  6  witli  the  direction  of  the  principal  stress, 
but  on  the  opposite  side  from  that  on  which  the  resultant  in 
Case  I  lies,  and  its  intensity  be  equal  to  that  of  either  prin- 
cipal stress. 

The  demonstration  of  Case  I  proves  this  principle  if 
Fig.  1  is  replaced  by  Fig.  2. 


FIG.  2. 

CASE  III.  Given  the  principal  stresses  of  the  same  kind 
but  having  unequal  intensities,  to  determine  the  intensity 
and  direction  of  the  resultant  stress  on  any  third  plane. 

Let  P  and  Q  be  compressive  and  the  intensity^  >  the 
intensity  q. 

The  following  identities  can  be  written: 


and 


4 


THEORY  OF  EARTH-PRESSURE. 


or  the  resultant  intensity  on  the  plane  CB  may  be  con- 
sidered as  being  the  resultant  of  two  intensities,  one  being 
the  intensity  of  the  resultant  stress  caused  by  two  like  prin- 
cipal stresses  having  the  same  intensity  \(p  -j-  q),  and  the 
other  the  intensity  of  the  resultant  stress  caused  by  two 
unlike  principal  stresses  having  the  same  intensity  %(p  —  q)- 


FIG.  3. 

The  intensity  of  the  resultant  stress  caused  by  the  first 
two  principal  stresses  will  be,  by  Case  I,  %(p  +  q),  and  the 
direction  of  the  resultant  will  be  normal  to  the  plane  CB. 
By  Case  II  the  resultant  of  the  second  pair  of  principal 
stresses  will  make  the  angle  0  with  the  direction  of  Py  and 
its  intensity  will  be  %(p  —  q)',  then  the  resultant  intensity 
can  be  .found  as  follows: 

In  Fig.  3  draw  MD  normal  to  EC,  and  make  LD  = 
J(j0  +  9f)j  with  L  as  a  centre  and  LD  as  radius,  describe 
an  arc  cutting  FD  at  F.  Then  the  angle  LFD  =  LDF  =  B. 
Lay  off  LG  —  \(p  —  q),  and  draw  GD,  which  is  the  result- 


THEORY  OF  EARTH-PRESSURE.  5 

iint  intensity,  and  the  intensity  of  the  resultant  stress  on 
OD  caused  by  the  two  principal  stresses  P  and  Q.  GD 
;;lso  represents  the  direction  of  the  resultant  stress  R. 

Since  the  intensities  of  the  principal  stresses  remain  con- 
stant, %(p  -\-  q)  and  \(p  —  q)  will  remain  the  same  for 
any  inclination  of  the  plane  GB\  hence  the  intensity  r  of 
the  resultant  depends  upon  the  angle  0  when  p  and  q  are 
given. 

From  Fig.  3, 

GL  cos  20  =  LM    and     GL  sin  20  =  GM, 
DM=DL  +  LM=$(p  +  q)  +  \(p  -  q)  cos  20, 


or 


r  =  Vp*  cos3  0  +  q*  sin2  0,   ....     (a) 


which  is  the  general  expression  for  the  intensity  of  the 
resultant  stress  of  a  pair  of  principal  stresses. 

As  the  angle  0  changes,  the  angle  ft  will  also  change, 
and  it  will  have  its  maximum  value  when  the  angle 
LGD  =  90°.  This  is  easily  proven  as  follows: 

With  L  as  centre  and  GL  as  radius  describe  an  arc; 
then  ft  will  have  its  maximum  value  when  the  line  DG  is 
tangent  to  the  arc;  but  when  DG  is  tangent  to  the  arc  the 
angle  LGD  is  a  right  angle,  since  LG  is'  the  radius  of  the 
arc. 

sin  max  ft  =  -  --  -,     .....     (b) 

from  which  the  following  can  be  easily  obtained: 
p       14-  sin  max  ft 

-*       -  '  ___  J_  (Ct 

q       1  —  sin  max  ft' 


6 


THEORY  OF 


which  expresses  the  limiting  ratio  of  the  intensities  of  the 
principal  stresses  consistent  with  equilibrium,  p  being 
greater  than  q. 

CASE  IV.  Given  the  intensity  and  direction  of  the  re- 
sultant stress  on  any  plane,  and  the  value  of  max  (3,  to 
determine  the  intensities  and  directions  of  the  principal 
stresses. 


FIG.  4. 


Let  ^4/)  represent  the  given  plane  and  GD  the  direction 
and  intensity  of  the  resultant  stress  at  the  point  D. 

Draw  DL  normal  to  AD,  and  draw  Dl,  making  the  angle 
max  ft  with  LD.  At  any  point  J  in  DL  describe  an  arc 
tangent  to  DI,  cutting  GD  in  K  and  draw  GL  parallel 
to  KJ\  with  L  as  a  centre  and  LG  as  radius  describe 


THEORY  OF  EARTH-PRESSURE.  7 

&  circumference.     This  circumference  will  pass  through  G 

C*  T 

and  be  tangent  to  DI\  hence  -777  =  sin  max  /?. 

L)  ±j 

ffl    _     Q 

Since  sin  max  /3  =  -  --  -  ,  and   GL  and  LD  are  com- 
p  +  q 

ponents  of  r, 

GL  =  l(p-q)     and     DL  =  \(p  +  q); 
then   ND  =  NL  +  LD  =  l(p-q)+i(p  +  q)=p, 
and      MD  =  LD  -  LM  =  l(p  +  q)  -  l(p  -  q)  =  q, 


which  completely  determines  the  intensities  of  the  principal 
stresses. 

According  to  Case  III,  the  direction  of  the  greater  prin- 
cipal stress  bisects  the  angle  between  the  prolongation  of 
LM  and  the  line  GL;  hence  RL  represents  the  direction 
of  the  greater  principal  stress,  and  that  of  the  other  is  at 
right  angles  to  RL. 

The  above  intensities  and  directions  being  determined, 
the  intensity  of  the  resultant  stress  on  any  other  plane 
passing  through  D  is  easily  determined  as  follows: 

Let  DY  represent  any  plane  passing  through  D,  draw 
DL'  normal  to  DY  and  equal  to  %(p  -f  q).  Draw  R'D 
parallel  to  RL,  and  with  L'  as  a  centre  and  L'D  as  radius 
describe  an  arc  cutting  R'D  at  0,  and  make  L'  G'  =  i(jP~~~Sr)5 
then  G'D  =  r'  =  the  intensity  of  the  resultant  stress  on 
DY. 

It  is  clear  that  if  the  value  of  max  fi  can  be  obtained 
for  a  mass  of  earth  that  the  construction  of  Fig.  3  can  be 
employed  in  determining  the  intensity  of  the  earth-pressure 
at  any  point  in  any  plane  within  the  mass. 


R  THEORY  OF  EARTH-PRfflSURfi. 

It  has  been  established  by  experiment  that  if  a  body  be 
placed  upon  a  plane,  that  (as  the  plane  is  made  to  incline 
to  the  horizontal)  at  some  angle  of  inclination  the  body 
will  commence  to  slide  down  the  plane,  and  that  this  angle 
depends  largely  upon  the  character  of  the  surfaces  in  con- 
tact. 

E 


FIG.  5. 

In  Fig.  5  let  AB  represent  a  plane  inclined  at  the  angle 
0  with  the  horizontal,  and  C  any  mass  just  on  the  point  of 
sliding  down  the  plane.  Let  EC  represent  the  weight  of 
the  mass  C,  and  ED  and  DC  the  components  respectively 
parallel  and  normal  to  the  plane  AB.  Then  DE  is  the 
force  required  to  just  keep  the  mass  C  from  sliding  down 
the  plane,  assuming  the  plane  to  be  perfectly  smooth,  or  if 
the  plane  is  rough  this  force  represents  the  effect  of  fric- 
tion. 

DE 


or  when  the  mass  C  is  about  to  slide,  the  resultant  pres- 
sure EC  on  AB  makes  the  angle  0  with  the  normal  to  the 


THEORY  OF  EARTH-PRESSURE.  0 

plane,  the  angle  0  being  the  inclination  of  the  plane  A  />, 
and  is  called  the  angle  of  friction. 

In  the  case  of  earth,  considered  as  a  dry  granular  mass, 
the  inclination  of  the  steepest  plane  upon  which  earth  will 
not  slide  is  called  the  angle  of  repose,  and  the  plane  the 
surface  of  repose. 

From  the  above,  then,  it  follows  that  in  a  mass  of  earth 
the  resultant  pressure  on  any  plane  cannot  make  an  angle 
with  the  normal  to  that  plane  which  is  greater  than  the 
angle  of  repose  0 ;  therefore  the  construction  of  Case  IV 
applies  to  earth  when  max  ft  is  replaced  by  <p.  The  values 
of  0  for  earth  under  various  conditions  are  given  in 
Table  II. 

The  preceding  principles  will  now  be  applied  in  deter- 
mining the  thrust  of  earth  against  a  retaining-wall. 


EARTH-PRESSURE. 

In  order  that  the  formulas  may  not  become  too  complex 
for  practical  use,  it  will  be  assumed  that  the  earth  is  a 
homogeneous  granular  mass  without  cohesion.  The  surface 
of  the  earth  will  be  considered  to  be  a  plane,  and  the  length 
of  the  mass  measured  normally  to  the  page  as  unity. 

*  Given  the  intensity  and  direction  of  the  resultant  stress 
at  any  point  in  any  plane  parallel  to  the  surface  of  the 
earth,  the  inclination  of  the  surface  of  the  earth  with  the 
horizontal,  and  the  angle  of  repose,  to  determine  the  in- 
tensity and  direction  of  the  resultant  stress  on  a  vertical 
plane  passing  through  the  same  point. 

*For  comparison,  see  the  "  Technic,"  1888;  a  construction  by 
Prof.  Greene. 

The  construction  follows  (see  Fig.  4,  above)  directly  from  Rau- 
kine's  Ellipse  of  Stress. 


10 


THEORY  OF  EARTH-PRESSURE. 


In  Fig.  6  let  BQ  represent  the  surface  of  the  earth,  and 
D  any  point  in  the  plane  AD  parallel  to  BQ  ;  draw  DQ 
normal  to  AD,  and  make  the  vertical  GD  equal  to  QD-, 
then  GD-y  is  the  intensity  of  the  resultant  pressure  atZX 
Draw  DM,  making  the  angle  0  with  LD,  and  with  L  as 
centre  describe  an  arc  tangent  to  DJ/and  passing  through 
G',  then  by  Case  IV  LG-y  =  ±(p  -  q),  LD-y  =  ±(p  +  q), 


FIG.  6. 


and  RL  bisecting  the  angle  QLG  is  the  direction  of  the 
greater  principal  stress.  To  determine  the  intensity  and 
direction  of  the  resultant  stress  at  D  on  a  vertical  plane, 
proceed  according  to  Case  IV.  Draw  R'D  parallel  to  RL 
and  DL'  =  DL  normal  to  DG.  With  L'  as  a  centre  and 
L'D  as  radius  describe  an  arc  cutting  R'D  at  R" 9  and  make 


THEORY  OF  EARTH-PRESSURE.  11 

L'G'  =  LG\    then  DG'  represents   the  direction  of   the 
resultant  stress,  and  DG'  *y  the  intensity  of  the  resultant. 
In  Fig.  6  the  angle  R'DL'  =  DR"L'  =  90°  -  GO  +  0. 
.'.   G'L'D  =  2w  -  26.     But  2V  =  GO  +  e;  hence   G'L'D 

=   GO  —   €. 

Draw  LY  —  LG  ;  then  the  angle  DLY  =  GO  —  e.  .:  Since 
LD  -  DL'  and  LY  =  LG  =  //#',  the  triangle  <7'Z//>> 
equals  the  triangle  LYD  and  the  angle  G'  DL'  —  e;  or  tf/ae 
direction  of  the  resultant  earth-pressure  against  a  vertical 
plane  is  parallel  to  the  surface  of  the  earth. 

From  Fig.  6, 

i(  P  ~  4)  cos  ^  =  &X  •  y> 


y  cos  e  — 

DY=DG'  =  DG- 
or 

/)#'  •  ^  =  Z>^-  y  —  (p  —  7)  cos  GO 

-  \(P  +  Q)  cos  e  -  {(p  -  q)  cos  GO, 
i(p-\-q):sm&)     ::     $(p  -  q)  :  sin  e, 
and 

P  +  9    - 
sin  to  =  -      -  sin  e, 

P-0 

or 


cos  «  =          - 


and  since  J(p  +  g')  sin  0  =  J(^  —  g), 

GO  =  -  —  -  ^cos2  e  —  cos2  0. 


cos 

sin 


12  THEORY  OF  EARTH-  PRESSURE. 

Substituting  this   value  for  cos  GO  in  the  equation  for 
DO'  .  y,  it  becomes 


V)  cos  e  -  i(p-q)  --    1/cos2  e  -  cos2  0, 


or  since  — 

sin  0      ^ 

DG'  -  y  —  J(p  -j-  <?)|cos  e  —  Vcos2  e  —  cos2  0J. 
In  a  similar  manner, 

DG  •  y  =  %(p  -j-  q)  {cos  e  -f  4/cos2  e  —  cos2  0}, 
and 


hence 


DG'  _  cos  e  —  Vcos8  e  —  cos2  0^ 
Z>6r        cos  e  -f-  '/cos2  e  —  cos2  0 


cos  e  -f  Vcos2  e  —  cos2  0 

Let  x  =  the  vertical  distance  between  the  two  planes 
BQ  and  A  D,  then 

/)#  =  DQ  =  x  cos  e. 

/ _\ 

r.^.  /  \    /  cos  e  —  4/cos2  e  —  cos2  0      A 

. '.  />£'  •  y  =  (x)  y\cos  e  -  .,    A 

cos  e  -j-  v  cos2  e  —  cos2  0 

which  is  the  expression  for  the  intensity  of  the  resultant 
earth-pressure  on  a  vertical  plane  at  any  depth  x  below  the 
surface. 
Let 


*  A  cos  e  —  Vcos2  e  —  cos2  0  ,  n 

*  ^4  =  cos  e  -  -—=  =.     .     .     (d) 

cos  e  -j-  '/cos2  e  —  cos2  0 

*  See  Rankine's  Applied  Mechanics  ;    Alexander's  Applied  Me- 
chanics j  Theories  of  Winkler  and  Mohr. 


THEORY  OF  EARTH-PRESSURE.  13 

The  average  intensity  of  the  resultant  earth-pressure  on 
a  vertical  plane  of  the  length  x  will  be 


ind  hence  the  total  pressure  will  be 

P  =  ~yA.  .    „ (e) 

Since  the  intensities  of  the  pressures  are  uniformly  varying 
from  the  surface,  and  increasing  as  x  increases,  the  appli- 
cation of  the  resultant  thrust  will  be  at  a  depth  of  \x  be- 
low the  surface. 

Considering  the  earth  as  an  unconfined  mass,  the  above 
formula  is  perfectly  general  and  can  be  applied  under  all 
conditions,  including  the  case  when  e  is  negative. 

The  resultant  stress  on  any  plane  as  AB,  Fig.  6,  can  be 
found  by  applying  the  principles  of  Case  IV.  Draw  PA 
parallel  to  RL,  make  AN  =  ZDand  NO  =  LG\  then  AO 
represents  the  direction  of  the  resultant  pressure  on  AB. 
Make  A C  =  AO',  then  the  area  of  the  triangle  ABC  mul- 
tiplied by  Y  is  the  total  pressure  on  the  plane  AB,  and  this 
pressure  is  applied  at  \AB  below  B. 

In  unconfined  earth  this  construction  is  perfectly  gen- 
eral and  applies  to  any  plane.  It  also  applies  equally  well 
to  curved  profiles.  An  example  illustrating  the  applica- 
tion of  the  method  will  be  given  in  the  applications.  See 
pages  22  and  23. 

The  following  graphical  construction,  Fig.  7,  is  more  con- 
venient than  that  of  Fig.  6. 

As  before,  let  BE  represent  the  surface  of  the  earth,  and 


14: 


THEORY  OF  EARTH- PRESSURE. 


A  D  a  plane  parallel  to  the  surface.  At  any  point  D  in 
tliis  plane,  draw  DE  vertical  and  make  DF  =  DE  \  draw 
FG  horizontal  and  make  the  angle  HFD  =  0. 

With  L  as  a  centre,  describe  an  arc  passing  through  G 
and  tangent  to  MF'9  then  with  L  as  a  centre  and  LF  as 


FIG.  7. 


radius,  describe  the  circumference  FON,  cutting  A  D  at  N; 
through  JV  draw  NO  parallel  to  AB,  then  draw  AC  nor- 
mal to  AB  and  equal  to  OG.  The  area  of  the  triangle 
ABC  multiplied  by  y  will  be  the  total  earth-pressure  on 
AB.  To  determine  the  direction  of  the  thrust  prolong  OG 
to  Q,  then  QN  is  the  direction  of  the  thrust. 

That  this  construction  is  equivalent  to  that  of  Fig.  6  is 


THEORY  OF  EARTH-PRESSURE.  15 

proved  as  follows.     The  triangle  GLF  of  Fig.  7  equals  the 
triangle  OLD  of  Fig.  6. 


q)      and     LF-y  ==  LO-y  == 
In  Fig.  6,  the  angle  JV^P  =  JVPJ.  =  90°-  i(oj-e)  -  «. 


In  Fig.  7,  the  angle  OLN=2e-2a.     But  GLN—  c^+e. 
.-.   ££0  =  w—.e  +  Za, 

and  £0  of  Fig.  7  equals  AO  of  Fig.  6. 

In  Fig.  7,  the  angle  QNO  =  90°  -  /?'. 

In  Fig.  6,  the  angle  OAB  =  90°  —  /?'. 

Therefore  the  direction  of  the  thrust  is  the  same  in  both 
constructions. 

The  two  constructions  given  above  are  all  that  is  re- 
quired to  determine  the  thrust  of  earth  upon  any  plane 
within  the  mass  of  earth,  as  one  can  be  used  as  a  check 
upon  the  other;  but  as  a  formula  is  often  very  convenient, 
a  general  formula  will  now  be  deduced  which  will  enable 
one  to  determine  the  values  of  E  and  d  for  any  plane  with- 
in a  mass  of  earth. 

GENERAL  FORMULA  FOR  THE  THRUST  OF  EARTH. 

In  Fig.  8,  let  BQ  represent  the  surface  of  the  earth  and 
A  B  any  plane  upon  which  the  earth-pressure  is  desired. 

Draw  AD  parallel  to  BQ  and  lot  the  vertical  distance 
QD  =  FA  =  x. 


16 


THEORY  OF  EARTH- PRESSURE. 


From  (e)  the  earth-pressure  upon  FA  is  parallel  to  the 
surface  and  equal  to 


FIG.  8. 


But  AF=  x  =  H(l  +  tau  a  tan  e)  =  //cos(6  ~  al. 


cos  a  cos  e 


«  cos   e  v  ' 


cos   «  cos   e 


Now   the   thrust  P   combined  with   the  weight  of  the 
prism  ABF  must  produce  the  resultant  pressure  upon  AB. 


THEORY  OF  EARTH-PRESSURE.  17 

Then  from  Fig.  8, 

H*y 

V  —  —^-  tan  a  (1  -f  tan  a  tan  e) 

A 

H*y  sin  a  cos  (e  —  a) 

~T~        cos2  a  cos  e      '     *     ^ 

^  =  |/(  F+P  sin  e)24-(P  cos  e)2  =  V  F2+Pa-j-  2  FP  sin  e. 
Substituting  (/)  and  (#)  in  this  it  becomes 
cos  (e  -  a) 


2    cos8  a  cos  e 


4/  sin2  a  -4-  2  sin  a  sin  e  cos  (e  —  a) }-  cos2  (e  — 

cos  e   ' 


cos-  e 
which  becomes,  by  replacing  A  by  its  value  from  (d  ), 


_ 
~" 


2     cos'1  a  cose 


+  sin2  a 


I  cos  e—  4/cos*  e—  cos2  0 

-j-  2  sm  <r  sm  e  cos  (e—  a)  — 

cos  e-[-  ^cos2  e—  cos'-*  0  ^     _     ^j 

(  cos  e  —  I/cos'2  e—  cos'2  0  )  2 
+  cos2  (e  -  a)  -J   -  -^-r 

(.  cos  e  -\-  ycos2  e—  cos-  0  ) 

\v  hich  is  the  general  equation  for  the  thrust  of  earth  upon 
any  plane  within  the  mass. 

To  determine  the  direction  of  the  thrust  of  the  earth, 
let  d  be  the  angle  which  the  direction  of  the  thrust  makes 
with  the  horizontal;  then,  from  Fig.  8, 


tan  d  —  -=-         -f  tan  e. 
P  cos  e 


18  THEORY  OF  EARTH-PRESSURE. 

Substituting  the  values  of  V  and  P  given  above,  this 
becomes 

tan  6  =  Sln  ^  cos  e  +  sin  e  cos  (e  -  a)  A 

cos  e  cos  (e  —  a)  A  v     ' 

where 

,  cos  e  —  l/cos2  e  —  cosa  0 

cos  e  -f  /cos2  e  —  cos2  0* 

Equations  (1)  and  (la)  are  readily  reduced  to  more  sim- 
ple forms  for  special  cases.  These  forms  will  be  found  on 
pages  23-25. 

Tlie  Plane  of  Rupture. — Although  it  is  not  necessary  to 
know  the  position  of  the  plane  of  rupture  in  order  to  deter- 
mine the  thrust  of  the  earth,  yet  it  may  be  of  interest  to 
know  its  position,  which  can  be  easily  determined  as  fol- 
lows : 

The  plane  of  rupture  will  be  back  of  the  wall  and  pass 
through  the  heel  of  the  wall.  The  resultant  earth-pressure 
will  make  the  angle  0  with  the  normal  to  this  plane.  Now 
the  tangent  of  the  angle  which  the  direction  of  the  result- 
ant earth-pressure  on  any  plane  makes  with  the  horizontal 
is  determined  from  the  formula 

«  sin  a 

tan  o  = -. -\-  tan  e. 

cos  (e  —  a)A 

If  GO  represents  the  angle  which  the  plane  of  rupture  makes 
with  the  vertical  passing  through  the  heel  of  the  wall, 
cf  =  GO  and  d  =  0  -f  GO. 

tan  (0  +  GO)  — — - — —  -f  tan  e, 

cos  (e  -  GO)  A 

from  which  the  value  of  GO  can  be  determined  for  any  case. 


THEORY  OF  EARTH-PRESSURE.  iu 

For  the  case  where  e  =  0,  e  being  positive  with  respect 
to-the  wall  and  negative  with  respect  to  the  plane  of  rupture, 
the  above  equation  becomes 

tan  (0  -f  GO)  =  --  —  r—  -  —  x  -  -7  —  tan  0, 
cos  (0  -f-  GO)  cos  0 

which  is  satisfied  when  GO  =  90°  —  0. 
For  the  case  where  e  =  0, 


,  sin 

tan  (0  +  G?)  =  - 


cos  a?  tan2 


o          0 


which  is  satisfied  when  G?  =  45°  —  — . 

& 

Reliability  of  the  Preceding  Theory. — The  preceding 
theory  is  based  upon  the  assumptions  that  the  earth  is  a 
homogeneous  mass  and  without  cohesion,  and  the  formulas 
are  deduced  under  the  assumption  that  the  surface  of  the 
earth  is  a  plane. 

All  writers  on  the  subject  have  considered  the  earth  as  a 
homogeneous  mass  and,  with  a  few  exceptions,  without 
cohesion. 

Old  and  recent  experiments  indicate  that  cohesion  has 
very  little  effect  upon  the  pressure  of  the  earth,  which  ex- 
phiins  why  it  has  not  been  considered  by  most  writers. 

The  assumption  of  a  plane  earth-surface  is  necessary 
whenever  practical  formulas  and  direct  graphical  construc- 
tions for  obtaining  the  thrust  of  the  earth  are  obtained. 
General  formulas  can  be  deduced  for  any  character  of  sur- 
face, but  they  are  too  complex  for  practical  use.  Those 
graphical  constructions  which  do  not  require  a  plane  earth- 


20  THEORY  OF  EARTH-PRESSURE. 

surface  are  not  direct  in  their  solution  of  the  problem,  but 
require  a  series  of  trials  to  obtain  the  maximum  thrust. 

If  the  earth-surface  is  not  a  plane,  one  can  be  assumed 
which  will  give  the  thrust  of  the  earth  sufficiently  exact 
for  all  practical  purposes. 

For  unconfined  earth  no  exceptions  can  be  taken  to  the 
preceding  theory,  the  assumptions  upon  which  it  is  based 
being  accepted,  and  for  confined  earth  the  theory  must  be 
true  when  the  direction  of  the  principal  stress  passing 
through  the  heel  of  the  wall  lies  entirely  within  the  earth. 

For  all  cases  in  which  a  and  e  are  positive  the  theories 
of  Rankine,  Winkler,  Weyrauch,  and  Molir  agree  and  give 
identical  results  with  the  preceding  theory,  as  they  should, 
being  founded  upon  the  same  assumptions. 

When  a  is  negative  Weyrauch  does  not  consider  his 
theory  reliable,  and  his  equations  lead  to  indeterminate  re- 
sults. 

Winkler  and  Molir  consider  their  theories  reliable  when- 
ever the  direction  of  the  principal  stress  passing  through 
the  heel  of  the  wall  lies  entirely  within  the  earth. 

Rankine's  method  of  considering  the  case  where  a  is 
negative  is  equivalent  to  assuming  that  the  introduction  of 
a  wall  does  not  affect  the  stresses  within  the  mass. 

It  may  be  concluded  that  the  preceding  theory  is  per- 
fectly exact  when  a  and  e  are  positive;  and  when  a  or  e  is 
negative  that  the  stresses  obtained  will  be  the  maximum 
which  under  any  circumstances  can  exist. 

For  the  case  where  e  is  negative  the  stress  obtained 
will  be  considerably  larger  than  the  actual  stress  (when 
a  wall  is  introduced),  depending  upon  the  magnitude 
of  e.  For  small  values  of  e  the  results  will  be  practically 
correct.  For  large  values  of  e  the  following  method  can 
be  employed  in  determining  the  thrust  of  the  earth.  The 


THEORY  OF  EARTH-PRESSURE. 


21 


method  depends  upon  the  assumption  that  the  pressure 
of  the  earth  is  normal  to  the  back  of  the  wall.  This  may 
or  may  not  be  the  case,  but  it  appears  to  be  the  most  con- 
sistent assumption  to  make  for  this  rare  and  not  important 
case. 


Fia.  9. 

*  In  Fig.  9,  let  AB  be  the  back  of  the  wall  and  Sfthe 
surface  of  the  earth.  Make  Ba  =  ab  =  ~bc  =  cd  =  etc. 
Some  prism  BAa  or  BAb  or  BAc,  etc.,  will  produce  the 
maximum  thrust  on  the  wall;  and  when  this  maximum 
thrust  is  produced,  the  resultant  pressure  on  the  plane  Aa 

*See  Van  Nostrand's  Magazine,  xvn,  1877,  p.  5.     "New  Con- 
structions in  Graphical  Statics,"  by  H.  T.  Eddy,  C.E.,  Ph.D. 


22  THEORY  OF  EARTH-PRESSURE. 

or  Ab  or  Ac,  etc.,  will  make  the  angle  0  with  the  normal 
to  the  plane. 

On  the  vertical  line  Ad'  lay  off  Aa'—ci'V  —  Vc',  etc.,  and 
draw  Aa"  making  the  angle  0  with  the  normal  to  Aa,  Ab" 
making  the  angle  0  with  the  normal  to  Ab,  etc.;  then  draw 
a' a",  b'b",  etc.,  perpendicular  to  AB,  and  draw  a  curve 
through  Aa",  b",  c",  etc.  Then  there  will  be  a  maximum 
distance  parallel  to  a' a"  between  Ad'  and  this  curve  which 
will  be  proportional  to  the  thrust  of  the  earth  against  AB. 
This  maximum  distance  multiplied  by  the  altitude  Ac  -f-  2 
and  the  product  by  y,  the  weight  of  a  cubic  foot  of  earth, 
will  be  the  pressure  of  the  earth. 

This  method  is  perfectly  general  and  can  be  applied  in 
any  case. 

If  the  earth-pressure  is  assumed  to  have  the  direction 
given  by  the  formulas  of  the  preceding  theory,  the  con- 
struction will  give  the  same  value  of  E,  the  pressure  of  the 
earth. 

Some  writers  assume  that  the  direction  of  E  makes  the 
angle  $>"—  <p  with  the  normal  to  the  back  of  the  wall  in 
all  cases.  This  assumption  cannot  be  correct  until  the  wall 
commences  to  tip  forward,  and  then  it  is  doubtful  that  such 
is  the  case  unless  the  earth  and  wall  are  perfectly  dry. 

To  be  on  the  side  of  safety  in  every  case,  it  is  better  to 
take  the  direction  of  E  as  given  by  the  above  theory. 

The  construction  of  Fig.  9  will  give  the  maximum  thrust 
for  any  assumed  direction  for  any  case. 


FOEMULAS  FOR  EARTH-PRESSURE. 

IN"  the  following  formulas  a  and  e  are  considered  as 
positive,  and  the  wall  is  assumed  to  be  one  foot  long. 

CASE  I.  General  case  of  inclined  earth-surface  and  in- 
dined  back  of  wall. 


2    cos2  a  cos  e 


/sin*  a  +  cos'  (e  -  a)  J  <**  *  -  j/cos«  C -cog^. 

(  cos  e  -f-  I/cos2  e  —  cos2 

.   j  cos  e  —  4/cos2  e  —  cos'2  0  )  ' 

-(-  2  sin  e  sin  a  cos  (e  —  a)  •<  -  — V 

(  cos  e  4-  I/cos2  e  —  cos2  0  ) 


or 


sin  «  cos  e  +  sin  e  cos  (e  —  a}A 

tan  tf  =  -  — ~-r -  — ;  .    (1«) 

cos  e  cos  (e  —  a)  A 

or  tan  d  = 7— —  -|-  tan  e,       ....   (1'a) 

cos  (e  —  a}A    ' 


L>4  FORMULAS  FOR  EARTH-PRESSURE. 

where 


cos  e  —  Vcos?  e  —  cos2  0 
A  =  cos  e  —  . 

cos  e  -+-  r  cos2  e  —  cos2  0 

CASE  II.  Surface  of  earth  inclined  and  a  =  0. 


/ 
cos  e  +  r  cos8  e  —  cos2  0 


(2) 


From  Diagram  I  the  values  of  A  can  be  found  for  all 
values  of  0  from  0°  to  90°  and  of  e  from  0°  to  90°,  vary- 
ing by  5°. 

6  =  e;  .......     (2a) 

or  for  all  vertical  walls  the  direction  of  the  earth-pressure 
is  parallel  to  the  surface  of  the  earth. 

CASE  III.  The  surface  of  the  earth  parallel  to  the  surface 
of  repose. 

6=0. 


E  _  I£y_  cos  (0  —  a)  ./sin2  a  +  cos2  (0  —  a)  ,^\ 

2    cos2  a  cos  0  ^   +  2  sin  or  sin  0  cos  (0  —  oi)' 

tan  rf  =  Bina  +  rin^B,^  -a)  _ 

COS  0  COS  (0  —  «) 

CASE  IV.  T7ie  surface  of  the  earth  parallel  to  the  surface 
of  repose  and  the  back  of  the  wall  vertical. 

e  =  0     and     a  —  0. 

^=^Zcos0 (4) 

d  =  0 .      (4rt) 


FORMULAS  FOR  EARTH-PRESSURE.  25 

CASE  V.  The  surface  of  the  earth  horizontal. 

E  =  —=¥•  y  tan2  a  +  tan4  ^45°  —  ^V      .       (5) 
tan  a 


tan  6  = 


—     — 
tan'  (45°-  f) 


CASE  VI.   The  surface  of  the  earth  horizontal  and  the 
back  of  the  ivall  vertical. 

e  =  0    and     a  =  0. 

(6) 

6  =  0  ...........     (6a) 

CASE  VII.  Fluid  pressure. 

e  =  <J>  =  0. 


$=a (7a) 

GRAPHICAL  CONSTRUCTIONS  FOR  DETERMINING  THE 
THRUST  OF  EARTH. 

The  following  constructions  are  perfectly  general,  and 
apply  to  any  plane  within  a  mass  of  earth.     When  applied 


26 


FORMULAS  FOR  EARTH-PRESSURE. 


for  determining  the  thrust  of  earth  against  a  retaining-wall, 
a  and  e  are  taken  as  positive. 

*  Construction  (a). 

Let  BE  represent  the  surface  of  the  earth  and  BA  the 
back  of  the  wall.  Draw  AF  parallel  to  BE;  and  at  any 
point  D  in  AF  lay  off  DF  equal  to  the  vertical  DE.  Draw 


FIG.  10. 

FG  horizontal,  and  FH,  making  the  angle  0  with  DF. 
With  any  point  J  in  DF  describe  the  arc  KI  tangent  to 
HF  at  /  cutting  FG  at  K,  and  draw  GL  parallel  to  KJ\ 
with  L  as  a  centre  and  LF  as  radius,  describe  the  circum- 
ference FQON  cutting  AD  at  N.  Through  J^draw  NO 

*  See  "Theorie  desErddruckes  auf  Grund  der  neueren  Anschau- 
ungen,"  by  Prof.  Weyrauch,  1881. 


FORMULAS  FOR  EARTH-PRESSURE.  27 

parallel  to  AB  cutting  the  circumference  FQON  at  0; 
at  A  draw  A  C  equal  to  OG  and  normal  to  AB;  the  area 
of  the  triangle  ABC  multiplied  by  y  will  be  the  thrust  of 
the  earth  on  the  wall. 

To  determine  the  direction  of  the  thrust  E,  prolong  OG 
to  ();  then  QN  will  be  the  direction  of  the  thrust. 

This  thrust  acts  on  the  wall  at  \AB  below  B. 

*  Construction  (b). 

Let  BQ  represent  the  surface  of  the  earth,  and  BA  the 
back  of  the  wall.  Draw  AD  parallel  to  BQ,  and  at  any 


FIG.  II. 

point  D  in  AD  draw  the  vertical  DG  equal  to  the  normal 
DQ\  draw  DM  making  the  angle  0  with  the  normal  DQ. 

*  This  construction  follows  directly  from   Rankine's  Ellipse  of 
Stress.     See  Rankiue's  Applied  Mechanics. 


28  FORMULAS  FOR  EARTH-PRESSURE. 

At  any  point  J  in  DQ  as  a  centre,  describe  the  arc  //T  tan- 
gent to  DM  cutting  DG  at  K,  and  draw  GL  parallel 
to  JK.  Bisect  the  angle  gZ^/and  at  ^4  draw  ^4P  parallel 
to  LR.  At  .4  draw  ^A^  normal  to  AB  and  equal  to  DL\ 
with  JV  as  a  centre  and  ^lA^  as  radius,  describe  an  arc 
AP  cutting  AP  at  P-  connect  P  and  N,  and  make  ^VO 
equal  to  LG\  with  ^4  as  a  centre  and  A  0  as  a  radius,  de- 
scribe the  arc  OC  cutting  AN  at  C ;  then  the  area  of  the 
triangle  ABC  multiplied  by  y  will  be  the  thrust  against 
the  wall.  The  direction  of  this  thrust  is  parallel  to  A  0 
and  it  is  applied  at  \AB  below  B. 

The  constructions  (a)  and  (b)  give  identical  results  in 
every  case. 

STABILITY  OF  TKAPEZOIDAL  WALLS 

As  the  majority  of  walls  retaining  earth  are  trapezoidal  in 
section,  the  stability  of  such  walls  alone  will  be  considered. 
If  other  forms  occur  in  practice  they  can  be  divided  into 
trapezoidal  sections  with  horizontal  beds,  and  the  stability 
of  each  considered,  commencing  with  the  upper  section. 

Walls  having  the  rear  faces  in  the  form  of  steps  can 
usually  be  considered  as  trapezoidal  in  section  by  re- 
placing the  stepped  portion  by  a  straight  line  which 
approximately  bisects  each  step.  If  the  front  faces  are 
stepped  they  can  be  treated  in  a  similar  manner. 

In  case  the  front  face  of  the  wall  is  curved  in  profile, 
the  curve  may  be  replaced  by  straight  lines  which  are 
chords  of  the  curve,  thus  binding  the  section  into  as  many 
trapezoids  as  there  are  chords. 

It  will  be  assumed  that  the  direction  and  magnitude  of 
the  earth-pressure  is  known,  that  the  position  and  extent 
of  the  back  of  the  wall,  and  the  width  of  the  top  are  given, 


FORMULAS  FOR  EARTH-PRESSURE. 


29 


to  determine  the  width  of  the  base  for  stability  against  over- 
turning, sliding,  and  crushing  of  the  material. 


Stability  against  Overturning.  —  Let  A  BCD,  Fig.  12,  rep- 
resent a  section  of  a  trapezoidal  wall,  TR  the  direction  of 
the  earth-thrust,  JG  the  vertical  passing  through  the  cen- 
tre of  gravity  of  the  wall,  and  JO  the  direction  of  the  re- 
sultant pressure  on  the  base  AD  caused  by  E  and  6r. 

As  long  as  R  cuts  the  base  AD,  the  wall  will  be  stable 
against  overturning.  When  R  takes  the  direction  JQ,  the 
wall  may  be  said  to  be  on  the  point  of  overturning;  then 

ON 

the  factor  of  safety  against  overturning  is  ~^.,  where  ON 


is  the  actual  value  of  E,  and  QNi\\Q  value  of  E  required  to 
make  the  resultant  R  pass  through  D. 

Stability  against  Sliding.  —  Since  the  wall  will  not  slide 


30 


FORMULAS  FOR  EARTH-PRESSURE. 


along  the  surface  DA  until  the  resultant  R  makes  an  angle 
with  the  normal  to  DA  greater  than  the  angle  of  friction 
0',  the  factor  of  safety  against  sliding  can  be  obtained  iis 
follows:  Draw  JP  making  the  angle  JMU '  =  0';  then 

PN 

the  factor  of  safety  against  sliding  is  --y^,  where  PJV is  the 

force  required  in  the  direction  of  E  to  make  R  make  the 
angle  0'  with  the  normal  to  AD,  and  ON  the  actual  value 
of  K 

Stability  against  the  Crushing  of  the  Material — In  ordi- 
nary practice  walls  for  retaining  earth  are  not  of  sufficient 
height  to  cause  very  large  pressures  at  their  bases,  but  it 
is  necessary  to  consider  the  subject  on  account  of  the  ten- 
dency of  the  bed-joints  to  open  under  certain  conditions. 


Let  AB,  Fig.  13,  represent  any  bed- joint  in  the  wall,  /' 
the  vertical  resultant  pressure  upon  the  joint,  and  .r0  the 
distance  of  the  point  of  application  from  the  centre  of  the 
joint. 

The  intensity  of  P  at  any  point  can  be  considered  as  com- 

p 

posed  of  a  uniform  intensity  pQ  =  -^-,anda  uniformly  vary- 
ing intensity  pj,  so  that  px  =  pQ  -f-  p0'.  Let  a  equal  the 
tangent  of  the  angle  CDE,  then  p9'  =  ax  and px  =^>0  +  ax. 


FORMULAS  FOR  EARTH-PRESSURE.  31 

The  pressure  upon  a  surface  (dx) — the  joint  being  con- 
sidered unity  in  the  dimension  normal  to  the  page — is 

pxdx  =  p0dz  4-  axdx, 
and  the  moment  of  this  about  DB  is 
(p0dx  -f  axdx)x. 
The  algebraic  sum  of  these  moments  for  values  of  x  be- 

T) 

tween  the  limits  ±  ~-  must  equal  Fx0,  or 

a 


Integrating, 

l2xnP 


0 
a  -  -r 


and 

or  making  x  =  -J5, 

/  /»     \  •»•) 

;;    ^ 

and  if  #,  be  replaced  by  \B  —  #,  where  §  is  the  distance 
from  A  to  the  point  where  P  cuts  the  base,  (Fig.  13,) 


and 


=  IB, 

p'  =  0     and    jti  =  2p,, 


32 


FORMULAS  FOR  EARTH-PRESSURE. 


from  which  it  is  seen  that  when  R  cuts  the  base  outside 
the  middle  third,  the  joint  will  have  a  tendency  to  open  at 
points  which  are  at  a  maximum  distance  from  R  where  it 
cuts  the  base. 

Therefore  in  no  case  should  the  resultant  pressure  be 
permitted  to  cut  the  base  outside  the  middle  third.  This 
makes  it  unnecessary  to  consider  the  stability  against  over- 
Burning. 


FIG.  14. 

Then  in  designing  a  wall  the  following  conditions  must 
exist  for  stability : 

I.  The  resultant  R  must  cut  the  base  for  stability  against 
overturning. 

II.  The  resultant  R  must  not  make  an  angle  vrith  the 
normal  to  the  base  of  the  ivall  greater  than  the  angle  of  fric- 
tion 0'. 


FORMULAS  FOR  EARTH-  PRESSURE.  33 

III.  The  resultant  R  must  not  cut  the  base  outside  of 
the  middle  third,  in  order  that  there  may  be  no  tendency  for 
the  bed-joints  to  open. 

The  above  three  conditions  apply  to  any  bed-  joint  of  the 
wall;  but  if  they  are  satisfied  at  the  base  and  the  wall  has 
the  section  shown  in  Fig.  14,  it  will  not  be  necessary  to 
consider  any  joints  above  the  base  unless  the  character  of 
the  stone  or  the  bonding  is  different. 

Determination  of  the  width  of  the  base  of  a  retaining- 
wall  under  the  condition  that  R  cuts  the  base  at  a  point 
rom  the  toe  of  the  wall. 

Let  H,  B',  x,  d,  and  E  be  given  to  determine  B. 

From  Fig.  14, 

KF=  f  sin  d  +  fcos  6  -^sin  d, 
666 


rrn  _  +  %BB'  -  Bx  -  ZB'x  -  B' 

-6(B  +  B') 


For  equilibrium 

E(KF)  =  G(HF)  =  B^B'  HW(ffF). 

A  , 

Substituting  the  values  of  A^and  HF'm  the  above  uii. 
reducing,  it  becomes 


=  %JL(H  cos  d  +  x  sin  6)  +  ZB'x  +  B'\    .     (8) 


34  FORMULAS  FOR   EARTH-PRESSURE. 

which  is  the  general  equation  for  the  width  of  the  base  of 
a  trapezoidal  wall. 

For  a  rectangular  wall  B'  =  B. 

For  a  triangular  wall  B'  =  0. 

For  a  wall  with  a  vertical  front  B'  -f-  x  =  B  or 
B'  =  B  -  x. 

For  a  wall  with  a  vertical  back  x  =  0. 

Equation  (8)  is  easily  transformed  to  satisfy  the  require- 
ments of  special  cases. 

.The  width  of  the  base  can  be  found  graphically  by  as- 
suming a  value  for  B  and  finding  the  value  of  Q-,  if  it  is 
less  than  %B  another  value  of  B  must  be  assumed,  and  so 
on  until  Q  is  equal  to  or  greater  than  ^B. 

FOKMULAS   FOR   TRAPEZOIDAL  AND   TRIANGULAR   WALLS. 

Formulas  for  the  width  of  the  base  of  trapezoidal  walls 
under  the  condition  that  the  resultant  R  cuts  the  base  at 
a  point  distant  from  the  toe  of  the  wall  equal  to  one  third 
the  width  of  the  base,  or  Q  —  %B. 

CASE  I.  Tlie  general  case  in  which  the  lack  of  the  ivall 
is  inclined,  and  E  makes  an  angle  with  the  horizontal. 


=  2*L  (ff  cos  d  +  x  sin  6\  +  ZB'x  +B'\  .     (8) 
CASE  II.   The  back  of  the  wall  vertical. 


=        cos  <?  +  £".        (9) 


FORMULAS  FOn  EARTH-PRESSURE. 


35 


CASE  III.   The  back  of  the  wall  vertical  and  the  thrust 
rma?  to  the  wall. 


=  0    and     3  =  0. 


(10) 


Q 


FIG.  15. 


If  B  =  Bf  and  x  —  0,  the  section  of  the  wall  is  a  rec- 
tangle, and  (9)  becomes 


and  (10)  becomes 


2jEJ         -  /n  , 

=  -^  cos  d,  .    .    .     (90) 

(10«) 


UNIVERSITY 

OF 


36  FORMULAS  FOR  EARTH-PRESSURE. 

Formulas  for  the  width  of  the  base  of  triangular  walls 
under  the  condition  that  the  resultant  R  cuts  the  base  at 
a  point  distant  from  the  toe  of  the  wall  equal  to  one  third 
the  width  of  the  base,  or  Q  —  %B. 

CASE  I.  The  general  case  in  which  the  back  of  the  ivall 
is  inclined,  and  E  makes  an  angle  with  the  horizontal. 

£>+£  (^sin  <J  -  z)  =  -^  (//cos  S  +  x  sin  6).     (11) 

CASE  II.   The  lack  of  the  wall  vertical. 
a  =  0. 


CASE  III.  The  back  of  the  wall  vertical,  and  the  thrust 
normal  to  the  wall. 

x  =  0     and     3  =  0. 

......     (13) 

The  above  formulas  do  not  contain  the  condition  that  I? 
shall  not  make  an  angle  greater  than  0'  ivith  the  normal  to 
the  base  of  the  wall. 

From  Fig.  15, 


which  expresses  the  condition  under  which  the  wall  will 
not  slide. 


FOUNDATIONS    FOR   WALLS   RETAIN- 
ING EARTH. 

The  design  of  the  foundations  for  retaining-walls  has 
received  but  little  attention  by  writers  upon  engineering 
subjects,  and  the  practical  engineer  has  not  published  to 
any  great  extent  examples  of  the  foundations  he  has  em- 
ployed under  the  countless  number  of  walls  erected  along 
railways,  highways,  canals,  etc. 

As  the  designing  of  foundations  resting  upon  earth,  for 
walls  retaining  earth,  introduces  several  features  which  do 
not  influence  the  ordinary  cases  of  foundations,  it  will  be 
best  to  make  a  special  investigation  for  such  conditions. 

The  intensity  of  the  foundation  pressure  upon  the  earth 
is  seldom  uniform,  due  principally  to  the  pressure  of  the 
earth  behind  the  wall  and  foundation  tending  to  overturn 
the  structure  as  a  whole ;  this  being  the  case,  evidently  the 
maximum  intensity  upon  the  earth  must  not  be  large 
enough  to  heave  the  earth,  and  the  minimum  intensity  must 
not  be  so  small  that  the  earth  may  heave  the  foundation. 

If  the  foundation  be  so  designed  that  neither  it  nor  the 
earth  can  be  heaved,  the  structure  may  yet  fail  by  sliding 
forward.  This  can  only  be  resisted  by  the  abutting  power 
of  the  earth  in  front  of  the  foundation  and  the  friction 
upon  the  base  of  the  foundation.  Usually,  however,  if 
there  is  no  danger  of  any  movement  in  a  vertical  plane, 
there  is  little  or  no  danger  of  any  movement  in  a  horizontal 
direction. 

As  in  any  structure  good  judgment  must  enter  into  the 
design,  the  formulas  which  will  be  demonstrated  must  be 

87 


38    FOUNDATIONS  FOR   WALLS  RETAINING  EARTH 


used  as  guides  only.  These  formulas  will  depend  upon  the 
angle  of  repose  0  of  a  homogeneous  granular  mass,  and  the 
specific  gravity  of  this  mass.  For  ordinary  earths  for  which 
the  weights  and  angles  of  repose  are  known  the  results  ob- 
tained hy  the  use  of  the  formulas  will  compare  very  favor- 
ably with  those  obtained  from  examples  of  the  best  practice. 
Depth  of  Foundations. — Given  the  angle  of  repose  0  of 
any  earth,  to  determine  the  depth  to  which  it  is  necessary 
to  sink  a  foundation  to  support  a  given  load.  The  surface 
of  the  earth  is  assumed  to  be  horizontal. 


q       **Y////////M\ 
FlO.     16. 

CASE  I.  When  the  intensity  of  the  pressure  on  the  base 
of  the  foundation  is  uniform. 

In  Fig.  16,  let  pQ  represent  the  intensity  of  the  pressure 
on  the  base  of  the  foundation. 

Now  when  the  masonry  is  about  to  sink  (see  Eq.  (c)), 


sin  0 

' 


1  —  sin 


-    nj 

'  ° 


1  —  sin  0 

_  ' 

1  +  sin  0' 


If  x'  represents  the  depth  to  which  the  foundation  .extends 
below  the  surface  of  the  earth  and  y  the  weight  of  a  cubic 


INUNDATIONS  FOR    WALLS  RETAINING  EARTH.     39 

foot  of  earth,  then  yx'  equals  the  vertical  intensity  of  the 
earth-pressure  on  a  plane  at  the  depth  of  the  lowest  point 
of  the  foundation. 

When  the  wall  is  on  the  point  of  sinking,  the  earth  must 
be  on  the  point  of  rising,  or 

_£     _  1  -f-  sin  0 
yx'  ~  1  —  sin  0' 
or 


In  any  case  />0  must  not  have  a  greater  value  than  that  ob- 
tained from  (15)  — 


=      tan' 


sm  y 

The  value  of  x'  as  obtained  from  (16)  is  the  least  allow- 
able value  consistent  with  equilibrium.  Since  x'  is  a  func- 

tion of  tan4  (45°—  —  j,  care  must  be  taken  that  0  is  assumed 

at  its  least  value.  As  0  becomes  smaller  the  value  of  x' 
increases  rapidly. 

CASE  II.  When  the  intensity  of  the  pressure  on  the  base 
is  uniformly  varying. 

Let  p  represent  the  maximum  intensity  of  the  pressure 
on  the  earth  and  p'  the  minimum  intensity;  then  for 
equilibrium  p  must  not  exceed  the  value  obtained  from  the 
following  equation  (see  15)  : 


—  sm 
For  any  assumed  depth  x'  the  maximum  value  of  p  can  be 


40    FOUNDATIONS  FOR    WALLS  RETAINING  EARTH. 

found  from  (17).  For  any  assumed  breadth  B"  of  the 
foundation  the  value  of  p  due  to  the  resultant  pressure  upon 
the  base  of  the  foundation  can  be  found  from  the  formulas 
on  page  31,  when  the  value  of  x0  has  been  determined;  this 
value  must  not  be  greater  than  the  value  of  p  found  from 
(17),  or  the  masonry  will  heave  the  earth. 

In  order  that  the  earth  may  not  heave  the  masonry,  p' 
must  not  be  less  than  the  value  obtained  from  the  following 
formula  : 


--  sm 
Then 


.....     (18) 


_  p  +  £  _  ^  C  /l  +  Bin0\«       /I  -  sin  0y  ) 
°  ~        a  2    Ul  -  sin  0/         U  +  sin  0/    J  ' 


which  expresses  the  maximum  value  p0  can  have  for  the 
equilibrium  of  the  earth  and  the  masonry. 

In  order  that  p'  may  never  be  less  than  the  value  obtained 
from  (18),  the  resultant  pressure  upon  the  base  of  the  foun- 
dation must  cut  the  base  within  a  certain  distance  of  its 
centre.  If  x0  be  this  distance,  then  (page  31) 


Substituting  the  value  of  p0  from  (19)  and  solving  for  #0, 


- 

6  ~' 


where 

A  +  riny     ^     y=/l-8in0y 

\1  —  sm     /  \1    -  sm    / 


*  Tabulated  values  of  X  and  Fare  given  on  page  72. 


FOUNDATIONS  FOR   WALLS  RETAINING  EARTH.     41 


Depth  of  foundations  when  the  surface  of  the  earth  has 
different  elevations  on  opposite  sides  of  the  structure. 


WALL 


FOUNDATION 

\P  ,P 


xy 


FIG.  17. 


This  case  is  illustrated  in  Fig.  17.     From  (17)  and  (18) 
for  equilibrium 


^ 

P    < 

f  ~ 


1  -f  sin  0  )  a 
1  —  sin  0  i 


(22) 


and 


Combining  (22)  and  (23)  in  the  value  of  p^ 

sin  0V 


„ 

' 


-  sn 


, 

-  ( 


Having  assumed  the  values  of  y  and  0  for  any  particular 
case,  the  above  formulas  determine  the  permissible  magni- 


4:2     FOUNDATIONS  FOR   WALLS  RETAINING  EARTH. 


tudes  of  the  intensities  at  the  heel  and  toe  of  the  founda- 
tion for  any  depth.  The  hreadth  of  the  ba^e  of  the 
foundation  may  now  be  assumed,  and  the  actual  intensities 
compared  with  those  permissible ;  if  p  is  too  large  or  p'  too 
small,  another  trial  must  be  made.  Usually  one  or  two 
trials  are  sufficient.  If  one  prefers  to  compute  the  width 
of  the  base  of  a  trapezoidal  foundation,  the  formula  given 
below  can  be  employed. 

Determination  of  the  breadth  B"  of  a  trapezoidal  foun- 
dation for  a  given  loading  and  a  maximum  intensity  p  at 
the  toe.  (Back  of  foundation  vertical. ) 


Fio.  18. 

Let  G  =  total  vertical  weight  supported  by  top  of  foun- 
dation ; 

E  =  thrust  of  earth ; 

p  =  maximum  intensity  of  pressure  at  toe  of  foun- 
dation as  found  from  (22) ; 
and     B"  =  breadth  of  base  of  foundation. 


FOUNDATIONS  FOR   WALLS  RETAINING  EARTH.     43 
Then 


-  G       tfMG      x'B*  W 


The  foundation  can  nearly  always  be  designed  as  a  trape- 
zoid  having  a  vertical  back,  and  then  if  necessary  the  batter 
in  front  can  be  stepped.  For  walls  under  twenty  feet  in 
height,  retaining  material  which  will  assume  a  slope  of  1£ 
to  1,  the  most  economical  foundation  is  rectangular  in 
section  if  the  base  must  be  four  feet  deep  to  escape  the 
action  of  frost.  Where  frost  need  not  be  considered,  of 
course  more  shallow  and  broader  foundations  can  be  em- 
ployed. 

Abutting  Power  of  Earth.  —  Let  the  surface  of  the  earth 
be  horizontal  and  the  body  pushing  the  earth  have  a  ver- 
tical face;  then  at  the  depth  x'  the  maximum  horizontal 
pressure  per  unit  of  area  is  (see  Case  I  above) 

1  -f-  sin  0 


and  since  q  varies  directly  as  #',  the  total  thrust  P  which 
the  earth  is  capable  of  resisting  is 

._  (x'Yr  1  +  sin  0 

2      1  -  sin  0*       '     '     '     (26J 

Bearing  Power  of  Earth.  —  The  bearing  power  or  the  in- 
tensity of  the  pressure  which  earth  can  resist  depends  not 
only  upon  the  character  of  the  earth,  but  upon  the  depth  to 
which  the  foundation  is  extended,  as  shown  by  the  formulas 
for  p  given  above.  For  example,  the  foundation  may  be 
very  broad  and  shallow  or  quite  narrow  and  deep.  The 


4:4:    FOUNDATIONS  FOR   WALLS  RETAINING  EARTH. 

intensity  of  the  pressure  in  the  first  case  being  considerably 
smaller  than  in  the  second,  and  both  conditions  fulfilling 
the  conditions  of  stability.  It  appears  then  that  the  bearing 
powers  of  earth  given  by  various  writers  must  be  employed 
with  caution,  unless  the  conditions  upon  which  the  values 
were  based  are  known. 


APPLICATIONS. 

The  determination  of  the  earth-pressure  by  the  pre- 
ceding formulas  and  graphical  constructions  is  a  very 
simple  operation  when  the  angle  0  has  been  determined  or 
assumed.  That  care  and  judgment  be  used  in  assuming 
the  value  of  0  is  very  important,  since  a  change  of  a  few 
degrees  in  the  value  of  0  sometimes  causes  a  large  change 
in  the  value  of  E.  An  inspection  of  Diagram  I  shows  that 
the  value  of  the  coefficient  A  increases  very  rapidly  as  0 
decreases. 

When  the  earth  to  be  retained  contains  springs,  the 
bank  must  be  thoroughly  drained  if  it  is  to  be  retained  by 
an  economical  tight  wall;  if  it  is  not  drained,  the  angle  0 
will  be  likely  to  become  very  small  as  the  earth  becomes 
wet. 

When  the  location  of  the  earth  to  be  retained  is  sub- 
jected to  jars,  the  value  of  0  will  be  decreased. 

Hence,  in  assuming  the  value  of  0,  the  engineer  must  be 
sure  that  the  value  assumed  will  be  the  least  value  which, 
in  his  judgment,  it  is  likely  to  have. 

In  constructing  the  wall  the  judgment  and  authority  of 
the  engineer  must  again  be  exercised  in  order  that  the  wall 
be  constructed  as  designed. 

In  all  cases,  to  insure  perfect  drainage  between  the  back 


FOUNDATIONS  FOR   WALLS  RETAINING  EARTH.     45 

of  the  wall  and  the  earth,  numerous  "  weep-holes"  should 
Be  provided  in  the  body  of  the  wall,  or  proper  arrange- 
ments made  to  carry  away  the  water  at  the  base  of  the  wall. 
To  facilitate  drainage,  the  backing  resting  against  the  wall 
should  be  sand  or  gravel. 

In  no  case  should  water  be  permitted  to  get  under  the 
foundation  of  the  wall,  neither  should  the  earth  in  front 
of  the  wall  be  allowed  to  become  wet. 

In  cold  localities  the  back  of  the  wall  near  the  top  should 
have  a  large  batter  to  prevent  the  frost  from  moving  the 
top  courses  of  stone.  As  a  guard  against  sliding,  the 
courses  of  the  wall  should  have  very  rough  beds.  The 
strength  of  a  wall  is  increased  the  nearer  it  approaches  a 
monolith. 

Care  should  be  taken  to  have  the  foundation  broad  and 
deep  enough  to  prevent  sliding  and  upheaving  of  the  earth 
in  front.  In  clay  the  foundation  should  be  deep,  while  in 
sand  or  gravel  it  may  be  broad  and  shallow. 

The  following  examples  illustrate  the  application  of  the 
formulas : 

Ex.  1.  Design  a  trapezoidal  wall  of  sandstone,  weighing 
150  Ibs.  per  cubic  foot,  having  a  width  of  3  ft.  on  top,  a 
height  of  30  ft.,  and  the  back  inclining  forward  5°,  to  re- 
tain a  bank  of  sand  sloping  upward  at  an  angle  of  20°. 

Data. 

y  -  100  Ibs.,  W=  150  Ibs.;  e  =  20°,  0  =  39°,  a  =  5°; 
H  =  30  ft.,  B'  =  3  ft.,  x  =  2.63  ft. 

1°.   Graphical  determination  of  the  values  of  E  and  6. 

The  graphical  solution  of  the  problem  is  shown  in  Fig.  19, 
where  E  is  found  to  equal  15,000  pounds.  $  lies  between. 
35°  and  36°. 


46     FOUNDATIONS  FOR    WALLS  RETAINING  EARTH. 
2°.  Algebraic  determination  of  E  and  £. 

E  = 


Fia.19. 


Substituting  the  values  of  B,  C,  D,  and  E  as  given  in  the 
tables,  and  that  of  A  as  given  by  Diagram  I,  this  becomes 


V(0.008)+(1.057)(0264)a+(0.061)0.264, 
tan  6  = ~    :— ^— ;  +  tan  e,      .    .     (I' a) 


-  45,000  (1.036)  V0.098  =  14,500  Ibs. 
sin  a 


cos  (e  —  a)  A 


FOUNDATIONS  FOR   WALLS  RETAINING  EARTH.     47 


tan  $  =  0.705  =  tan  35°  11',  about. 

3°.  Algebraic  determination  of  the  value  of  £  under  the 
assumption  that  Q  =  %B. 


|  Hcos  S  +  a-sin  o  j  +  ZS'x  +  £".    .     (8) 


0  X  0.817  +  2.63  X  0.576J  +  6  X  2.63  +  9, 

oU  X  loU 

B*  +  7.795  =  172.53, 

B  =  -  3.89  ±  A/172.53 
.*.  B  =  13.69  -  3.89  =  9.80  ft.; 

or,  practically,  10  feet  is  the  required  width  of  the  base. 

4°.  To  determine  if  the  wall  will  slide  on  a  foundation  of 
sandstone. 

From  (14), 

.  E  cos  8 


Taking  B  =  10  ft.,    G  =      --3°X150  =  29250  llis. 


4:8    FOUNDATIONS  FOR   WALLS  RETAINING  EARTH. 
ti  =  35°  11',  cos  6  =  0.817,  and  sin  6  =  0.576,  then 


14500  X  0.817 
G  -f  E  sin  tf  ~~  29250  +  14500  X  0.576  " 

From  Table  II,  the  value  of  tan  0'  for  masonry  is  0.6  to 
0.7;  hence  there  is  no  danger  of  the  wall  sliding  on  the 
foundation. 

According  to  the  Engineering  News  formula  the  base  of 
this  wall  would  be  fZT"plus  a  few  inches  for  good  luck," 
or  about  13  feet,  and  by  the  old  rule  of  one  third  the  height 
10  feet. 

Ex.  2.  Design  a  trapezoidal  wall  of  sandstone  weighing 
150  Ibs.  per  cubic  foot,  having  a  width  of  3  ft.  on  top,  a 
height  of  30  ft.,  and  the  back  inclining  backward  15°,  to 
retain  a  bank  of  sand  sloping  upward  at  an  angle  of  30°. 

Data. 

y  =  100  Ibs.,  W=  150  Ibs.  ;  e  =  30°,  0  =  33°,  a  =-  15°; 

H  =  30  ft.,  B'  =  3  ft.,  x  =  8  ft. 

1°.  Graphical  determination  of  the  values  of  E  and  3. 

In  Fig.  19,  let  B  6r  rep-resent  the  surface  of  the  earth,  and 
AB  the  back  of  the  wall.  Draw  AF  parallel  to  BG,  and 
from  any  point  D'  in  AF  lay  off  D'  F  equal  to  the  vertical 
D'G,  and  draw  FL  horizontal;  lay  off  the  angle  IFD'  =  <p 
=  33°,  and  locate  the  point  M  in  D'Fso  that  if  an  arc  be 
described  with  M  as  a  centre  and  L  M  as  a  radius  the  arc 
will  be  tangent  to  IF',  then  with  M  as  a  centre  and  MF  as 
a  radius,  describe  the  circumference  FHJ  and  draw  /// 


FOUNDATIONS  FOR   WALLS  RETAINING  EARTH.     4^ 

parallel  to  'AB\  at  A  draw  AL  perpendicular  to  AB  and 
equal  to  HI.     Then 


—  r  = 


=  14800  =-.  E. 


To  determine  <5,  prolong  HI  to  A"  and  draw  KJ.  Then 
the  angle  which  this  line  makes  with  the  horizontal  is 
equal  to  6,  which  is  6°  to  7°  in  this  case. 


FIG.  20. 


2°.  Algebraic  determination  of  E  and  d. 

Substituting  in  (1)  and  remembering  that  a  is  negative, 


E  =  45000  (0.875)  Vo.067  +  0.183  -  0.111  =  14600  Ibs. 


50    FOUNDATIONS  FOR   WALLS  RETAINING  EARTH. 

From  (I'  a), 

—  0  25Q 

tan  *  =  "577  =  -  °'133  =  tan  -  ?0' 


3°.  Algebraic  determination  of  the  value  of  B  under  the 
assumption  that  Q  =  %B. 

Substituting  the  proper  values  in  (8)  and  remembering 
that  a  is  negative, 


B  =  -  4.7  ±  V163.44  +  (4.7)'  =  9.0  ft. 

Ex.  3.  Determine  the  dimensions  of  a  brick  wall  hav- 
ing a  vertical  back  to  retain  a  bank  of  sand  sloping  up- 
ward at  an  angle  of  20°.  (f>  =  30°,  H=  20',  B'  =  2', 
y  =  100. 

1°.  Algebraic  determination  of  E  and  d. 

Since  a  =  0, 


(2) 


E  =  i00><100  ot424  =  8480;  say,  8500  Ibs. 
A 

The  value  of  A  is  readily  found  from  Diagram  I. 
d  =  e  =  20°,     since     a  =  0. 

2.  Algebraic  determination  of  the  value  of  B  under  the 
condition  that  Q  =  %B. 


(9) 


FOUNDATIONS  FOR    WALLS  RETAINING  EARTH.     51 
From  Table  I,  W  =  125  Ibs.     Then 


or  B*  -f  6.655  =  131.84. 

B  =  _  3.33  ±  -v/131.84  +"03*, 
and 

B=  —  3.33  +  11.94  =  8.61  ft. 

Ex.  4.   Determine  the  value  of  B  in  Ex.  3  under  the 
assumption  that  e  =  0  (horizontal  earth-surface). 


2  2  y       1    -  sin 


or  #  =  20000  (0.333)  =  6666,  say  6700  Ibs. 
Since  a  =  0,  and  e  =  0,  d  =  0, 


(10) 


JSa  +  2£  =  111.2; 
B  =  -1  ±  Vlll.2  +  1, 
and  B  =  -1  +  10.59  =  9.6  ft. 

Ex.  5.   Determine  the  value  of  B  in  Ex.  3,  under  the 

assumption  that  e  =  0  =  30°. 

^  =  ^^cos  0  =  20000  (0.866)  =  17320  Ibs. 
From  (9), 


52    FOUNDATIONS  FOR   WALLS  RETAINING  EARTH. 

B*  +  15.86#  =  244.05; 

B  =  -  7.93  +  ^244.05  +  Of*, 
and      B  =  -  7.93  -f  17.52  =  9.6  ft. 

Ex.  6.  Determine  the  resultant  pressure  against  the 
back  of  a  wall  when  the  surface  of  the  earth  carries  a 
load  equivalent  to  5  feet  in  depth  of  sand. 

H  =  30  ft.,  a  =  10°,  0  =  30°,  e  =  0,  and  y  =  100 
Ibs. 


FIG.  21. 

Graphical  solution  of  the  problem. — In  Fig.  21,  let  .B/tf 
represent  the  surface  of  the  earth,  and  BA  the  back  of 
the  wall. 

Make  ST=  5,  and  draw  HT  and  BH.  Draw  AR  par- 
allel to  BS,  parallel  to  HT,  and  make  LR  equal  to  Z77; 
lay  off  the  angle  LRP  equal  to  30°;  with  0  as  a  centre 


FOUNDATIONS  FOR   WALLS  RETAINING  EARTH.     53 

draw  an  arc  passing  through  L  tangent  to  PR,  and  then 
with  OR  as  a  radius  describe  the  circumference  of  the 
circle  RQM,  and  at  M  draw  MN  parallel  to  AH  \  at  A 
and  normal  to  A  H  draw  A  0  equal  to  NL.  Then 


The  direction  of  E  will  be  parallel  to  QM. 

To  determine  the  point  of  application  of  E,  find  the 
centre  of  gravity  E'  of  A  B  VC,  and  draw  E'D  parallel  to 
A  0,  then  D  will  be  the  point  of  application  of  E. 

E'  can  be  found  as  follows:  Produce  A  C  and  BV,  make 
AI=  CK=BV,BG  =  VF=  J(7,aud  join  /*  and  /  and 
G  and  K.  Then  E',  the  intersection  of  FI  and  GK,  will 
be  the  centre  of  gravity  of  AB  VC.  BD  can  be  found 
from  the  formula 

BD  c      L0°  -  l 

~  3 

Ex.  7.  Determine  graphically  the  value  of  E  when  e  =  0 
and  a  =  0,  0,  y,  and  H  being  given. 

In  Fig.  22  let  BF  represent  the  surface  of  the  earth,  and 
AB  the  back  of  the  wall.  Draw  AL  parallel  to  BF  and 
make  IL  —  IF;  lay  off  the  angle  GLH  =  0,  and  at  .any 
point  K  in  LH  draw.  JOT  perpendicular  to  /TL,  and  lay 
off  Jf  0  =  JOT;  draw  W  parallel  to  01.  Then  will  the  arc 
IN,  described  with  J  as  a  centre  and  //as  a  radius,  pass 
through  /and  be  tangent  to  6r/;  with  /  as  a  centre  and 
//  as  radius  describe  the  circumference  LH  ;  at  A  lay  off 
AC—HI  and  normal  to  AB.  Then 


54    FOUNDATIONS  FOR   WALLS  RETAJNINO  EARTH. 

E  is  parallel  to  BF  and  applied  at  D,  AD  being  equal  to 
\AB. 


FIG.  23. 

Ex.  8.  Determine  the  earth-thrust  on  the  profile  shown 
in  Fig.  23,  H,  y,  <j>9  and  e  being  given. 

Graphical  solution  of  the  problem.  —  Let  BCDEA  repre- 
sent the  given  profile,  and  let  the  surface  of  the  earth 
be  horizontal.  Prolong  BC  until  it  intersects  8  A  in1  8; 
draw  SR  normal  to  BCS  and  equal  to  the  intensity  of  the 
earth-pressure  at  S;  connect  B  and  R.  Then  from  the 
middle  point  of  B  C  draw  GF  parallel  to  SR;  the  distance 
GF  multiplied  by  y  will  be  the  average  intensity  of  the 
earth-pressure  on  BC.  In  a  similar  manner  the  average 
intensities  on  CD,  DE,  and  EA  can  be  found,  and  hence 
the  total  pressures  on  each  determined.  The  points  of  ap- 


plication of  these  resultant  pressures,  El 


z  , 


and 


FOUNDATIONS  FOR   WALLS  RETAINING  EARTH.     55 

can  be  found  by  the  method  used  in  Ex.  6  for  finding 
the  centre  of  gravity  of  a  trapezoid.     The   directions  of 
B 


FIG.  23. 

j?, ,  EI ,  E^ ,  and  Et  are  found  from  the  construction  on  the 
right. 

Ex.  9.  Determine  the  thrust  of  the  earth  against  a  ver- 
tical wall  when  e  is  negative. 

For  the  explanation  of  this  construction,  see  page  21, 
Fig.  9. 

Ex.  10.  From  the  following  data  determine  E,  d,  and 

Q- 

e  =  0,  0  =  38°,  a  =  10°  23';  y  =  90  Ibs.,  W  =  170  Ibs.j 

H  =  15  ft.,     B  =  6  ft.,     B'  =  2  ft. 
Ans.  E  -  3037  Ibs.,  6  =  37°  37',  Q  =  2.2  ft. 

Ex.  11.  Determine  the  dimensions  of  a  trapezoidal  wal] 
built  of  dry,  rough  granite,  having  a  vertical  back  and 
being  20  feet  high,  to  safely  retain  the  side  of  a  sand  cut, 


50    FOUNDATIONS  FOR   WALLS  RETAINING  EARTH. 

the  surface  of  the  sand  being  level  with  the  top  of  the  wall. 
jr=lG5.1bs.,  y=lOO  Ibs.,  0=:33°  40',  H  =  20  ft., 
B'  =  2  ft. 

.4w*.  ^  =5734  Ibs.,  tf  =  0,  5  =  8  ft.,  and   Q  =  2.8  ft., 
about. 

Ex.  12.   The  same  as  Ex.  11,  with  a  —  8°  instead    of 
«  =  0.    ' 

Ans.  E  —  6330  Ibs.,  B  =  9.98  ft.,  and  Q  =  2.7  ft. 


FIG.  34. 

Ex.  13.  What  must  be  the  dimensions  of  a  rubble  wall 
of  large  blocks  of  limestone,  laid  dry,  to  retain  a  sand 
filling  which  supports  two  lines  of  standard-gauge  railroad 
track  ?  (Assume  the  depth  of  sand  to  produce  a  pressure 
on  the  earth  equal  to  that  produced  by  the  railroad  and 
trains  as  4  feet.) 


FOUNDATIONS  FOR    WALLS  RETAINING  EARTH.     57 

H  =  15  ft.,  a  =  8°,  0  =  33°  40',  y  =  100  Ibs.,  W  =  170 
Ibs.,  B'  =  3.5  ft. 

^?w.   ,#=5760  Ibs.,  tf  =  26°  7',  5  =  8  ft,  §  =  2.7ft. 
Ex.   14.  Determine    E,  d,  B,   and    ft  when    W  =  170 
Ibs.,  y  =  100  Ibs.,  or  =  8°,  e  =  0  =  33°   40',  H  =  20  ft., 
B'=  2ft. 
.4  //«<?.  E  —  21760  Ibs.,  tf  ==  32°  25',  .5  =  9  ft.,  Q  =  3  f  t. 

*  Ex.  15.  A  wall  9  ft.  high  faces  the  steepest  declivity 
of  earth  at  a  slope  of  20°  to  the  horizon;  weight  of  earth 
130  Ibs.  per  cubic  foot,  angle  of  repose  30°.     Determine 
E  when  a  —  0. 

Ans.  E  =  2187  Ibs. 

*  Ex.  16.  e  =  33°  42',  0  =  36°,  H  =  3  ft.,  y  =  120 
Ibs.,  a  =  0.     Determine  E. 

Ans.  E  =  278  Ibs. 

*  Ex.  17.  0  =  25°,  6=0,  «  =  0,  H=4  ft.,  y  =  120 


Ans.  E  =  390  Ibs. 

*  Ex.  18.   0  =  38°,   €  =  0,    a  =  0,  ti  -3  ft.,   y  =  94 
Ibs.,  E  =  ? 

Ans.  E  =  100.5  Ibs. 

*  Ex.  19.  A  ditch  6  feet  deep  is  cut  with  vertical  faces 
in  clay.     These  are  shored  up  with  boards,  a  strut  being 
put  across  from  board  to  board  2  feet  from  bottom,  at 
intervals  of  5  feet  apart.     The  coefficient  of  friction  of 
the  moist  clay  is  0.287,  and  its  weight  120  Ibs.  per  cubic 
foot.     Find  the  thrust  on  a  strut,  also  find  the  greatest 
thrust  which  might  be  put  upon  the  struts  before  the  ad- 
joining earth  would  heave  up. 

Ans.  E  =    1230  Ibs. 
Thrust  per  strut  =    6128  Ibs. 
Greatest  thrust    =  19029  Ibs. 
*  Alexander's  Applied  Mechanics. 


58    FOUNDATIONS  FOR   WALLS  RETAINING  EARTH. 

Ex.  20.  Examine  the  stability  of  the  wall  shown  in  Fig. 
25,  and  design  a  foundation  which  will  be  safe  as  long  as 
the  condition  of  the  earth  remains  unchanged ;  the  weight 
of  the  masonry  being  145  pounds  per  cubic  foot,  that  of 
earth  100  pounds,  and  the  angle  of  repose  of  the  earth  such 
that  it  will  stand  at  a  slope  of  1|  to  1. 

Stability  of  the  Wall  upon  the  Foundation. — Replacing 
the  stepped  back  by  the  line  BD,  the  thrust  of  the  earth 
is  found  to  be  about  9900  pounds.  The  direction  of  this 
force  is  shown  in  Fig.  25 ;  since  it  cuts  the  base  of  the  wall 
there  is  no  danger  of  the  structure  being  overturned,  how- 
ever large  E  may  become. 


Determining  the  centre  of  gravity  of  the  wall  and  also  its 
weight,  and  combining  this  with  E,  the  resultant  pres- 
sure upon  the  base  of  the  wall  is  found  to  be  about  32,000 
pounds.  This  resultant  makes  an  angle  of  less  than  11 


FOUNDATIONS  FOR    WALLS  RETAINING  EARTH.     59 

degrees  with  the  normal  to  the  base.  Now  since  for  masonry 
sliding  upon  masonry  the  angle  of  friction  is  from  31  to  35 
degrees  (Table  II),  there  is  no  danger  of  failure  by  sliding 
upon  the  foundation.  Since  the  resultant  cuts  the  base 
within  the  middle  third  the  entire  base  is  subjected  to  com- 
pression, and  there  will  be  no  tendency  for  the  joints  to 
open  at  the  heel. 

Failure  by  the  crushing  of  the  material  need  not  be  con- 
sidered, as  the  maximum  intensity  of  the  pressure  upon  the 
base  is  many  times  smaller  than  the  ultimate  strength  of 
the  material.  See  page  68. 

The  resultant  pressure  upon  the  base  can  be  found  also 
by  assuming  the  earth  on  the  left  of  the  vertical  to  be  sup- 
ported by  the  wall,  and  that  the  pressure  of  the  earth  upon 
the  right  of  this  line  acts  against  the  vertical  plane  KD ; 
this  pressure  is  about  5800  pounds,  and  is  horizontal.  Com- 
bining this  force  with  the  weight  of  the  wall  and  earth  on 
the  left  of  the  line  KD,  the  resultant  pressure  upon  the 
base  is  found  to  be  the  same  in  magnitude  and  direction  as 
by  the  first  method. 

The  Foundation. — The  depth  of  the  foundation  must  be 
below  the  action  of  frost;  let  this  be  assumed  as  5  feet; 
then  by  (22),  with  x'  =  5  feet,  the  maximum  allowable  pres- 
sure at  the  toe  of  the  foundation  is  about  6000  pounds  per 
square  foot,  and  by  (23)  the  minimum  allowable  pressure  is 
about  200  pounds  for  x"  =  25  feet. 

Assuming  that  the  foundation  is  vertical  at  the  back  and 
trapezoidal  in  section,  the  length  of  the  base  B"  can  be 
found  from  (25),  which  will  satisfy  the  condition  of  maxi- 
.  mum  pressure  at  the  toe.  Letting  p  =  5000  and  x'  =  5, 
and  solving  (25),  B"  is  found  to  be  between  12  and  13  feet; 
say  13  feet. 

To  determine  if  this  width  is  sufficient  to  satisfy  all  the 


60     FOUNDATIONS  FOR    WALLS  RETAINING  EARTH. 

conditions  of  equilibrium,  the  resultant  of  all  forces  acting 
upon  the  base  must  be  found. 

*  The  total  earth-pressure  upon  the  vertical  HK  is  about 
8900  pounds.  Combining  this  with  the  weight  of  the  wall, 
earth  supported  by  the  wall,  and  that  of  the  foundation,  the 
resultant  vertical  pressure  is  found  to  be  about  40,600 
pounds,  and  is  applied  within  the  middle  third  of  the 
base,  about  1.7  feet  to  the  left  of  the  centre. 

The  intensity  of  the  pressure  at  the  toe  is  (page  31) 


p  =     i  +  =  about  5600  pounds, 

(  lo      )      lo 

which  is  less  than  the  maximum  allowable  intensity.  The 
intensity  at  the  heel  isp'  =  %p0  —  p  =  about  600  pounds, 
which  is  greater  than  the  minimum  allowable  intensity; 
hence  this  foundation  is  sufficient  to  prevent  settlement. 

A  glance  at  Fig.  25  is  sufficient  to  show  that  the  foun- 
dation will  not  slide  upon  the  earth  even  if  the  movement 
were  not  opposed  by  a  force  of  some  4000  pounds,  being  the 
abutting  power  of  the  earth  in  front  of  the  foundation. 

The  above  foundation  then  fulfils  all  the  conditions  of 
stability,  but  to  allow  for  contingencies  the  foundation 
should  be  designed  under  the  assumption  that  0  may  be 
somewhat  smaller  than  its  average  value,  which  is  equivalent 
to  broadening  the  base  if  the  depth  remains  the  same 

*  The  pressure  against  the  foundation  in  front  of  the  wall  has 
been  neglected,  but  can  be  easily  included  by  taking  the 
instead  of  KMP. 


PROFILES  OF  WALLS  RETAINING  EARTH.        61 
EXAMPLES  OF  RETAINING-WALL  PROFILES. 


r 


FIG.  26. 


.      . 

A  Standard  Profile  used  for  the  past  twenty  years  near  New  York 
City,  where  railway  tracks  have  been  lowered  below  the  streets. 
(Engineering  News,  1889.) 


FIG.  27. 


Profile  of  Retaining-wall  at  Ferdinand  Street  Bridge.  "Boston,  Mass. 
(City  Engineer's  Report,  1891  ) 


62 


PROFILES  OF  WALLS  RETAINING  EARTH. 


FIG.  28. 


Profile  of  Abutment  at  Ferdinand  Street  Bridge,  Boston,  Mass 
(City  Engineer's  Report,  1891.) 


FIG.  29. 

Profile  of  Retaining-wall  at  Boylston   Street  Bridge,  Boston,  Mass 
(City  Engineer's  Report,  1883.) 


PROFILES  OF  WALLS  RETAINING  EARTH.          63 


FIG.  30. 


Profile  of  Retaining-wall  at  Liverpool,  England.     (Harcourt). 


FIG.  31. 

Profile  of  Retaining-wall,  Thames   Embankment,    Chelsea.     (Uar- 

court.} 


PROFILES  OF  WALLS  RETAINING  EARTH. 


FIG.  32. 

Profile  of  Retaining-wall    Thames  Embankment,  Lambeth.     (Ear 

court.) 


i+ 216 * 

FIG.  33. 
Profile  of  Concrete  Retaining-wall  at  Chatham.     (Harcourt.) 


PROFILES  OF  WALLS  RETAINING  EARTH.          65 


FIG.  34. 
Profile  of  Retaining-wall  at  Millwall.     (Harcourt.) 


FOUNDATIONS. 

The  proper  proportions  of  foundations  to  suit  different 
conditions  have  been  the  results  of  experience  principally, 
though  theory  enters  into  their  design  in  many  ways. 
Under  certain  logical  assumptions,  the  offsets  of  wood,  iron, 
or  stone  foundation  courses  can  be  as  accurately  determined 
as  the  stresses  in  any  beam  subjected  to  cross-bending. 
The  strengths  of  various  materials  which  enter  into  the 
construction  of  foundations  have  been  fairly  well  determined 
experimentally,  so  that  the  allowable  intensities  of  the 
pressures,  and  consequently  the  areas  of  the  foundation 
courses,  can  be  accurately  determined.  There  remains  the 
most  difficult  portion  to  be  decided,  namely,  the  proper 
intensity  of  the  pressure  upon  the  earth  which  must  sup- 
port the  load.  Under  certain  assumptions  this  can  be 
computed,  but  the  best  of  judgment  must  be  exercised  in 
making  the  assumptions  upon  which  calculations  are  based. 

Whenever  possible,  the  intensity  of  the  pressure  upon  the 
earth  should  be  uniform  under  all  parts  of  the  structure 
(assuming  the  earth  to  be  homogeneous)  ,  and  the  founda- 
tions extend  to  the  same  depth.  Theoretically,  a  greater 
intensity  is  allowable  at  a  greater  depth,  but  practically  this 
may  lead  to  unequal  settlement,  due  to  the  compressibility 
of  the  earth,  which  theory  does  not  take  into  account. 

FOUNDATIONS  utoN  ROCK. 


In  preparing  a  bed  for  the  structure  to  be  erected  all  loose 
and  decayed  parts  of  the  rock  must  be  removed,  and  the- 
surf  ace  made  as  nearly  horizontal  as  practicable  ;  when  the 
surface  is  inclined,  it  may  be  cut  into  steps  with  horizontal 

66 


FOUNDATIONS   UPON  ROCK.  67 

and  vertical  faces ;  if  holes  exist,  they  may  be  filled  with 
concrete.  In  some  cases  a  proper  surface  for  supporting  the 
proposed  structure  can  be  secured  by  covering  the  rock 
surface  with  a  layer  of  concrete,  which  may  vary  from  a 
few  inches  to  two  or  more  feet  -in  thickness.  (Figs.  39 
and  42.) 

The  maximum  intensity  of  the  pressure  upon  a  rock  foun- 
dation should  not  exceed  one  sixth  the  crushing  strength  of 
the  rock  for  a  steady  and  uniform  load,  or  one  tenth  the 
crushing  strength  for  a  load  due  to  the  weight  of  the  struc- 
ture plus  a  varying  load  such  as  is  caused  by  wind  or  earth 
pressure. 

In  no  case  should  any  portion  of  the  horizontal  joints  be 
subjected  to  tension.  The  maximum  deviation  of  the 
centre  of  pressure  from  the  centre  of  gravity  of  the  base 
section,  when  the  section  is  a  symmetrical  figure,  can  be 
found  from  the  formula 

Xo==Ay'  (Rankine)5 

where  x0  =  the  maximum  deviation  sought; 

/  =  the  moment  of  inertia  of  the  section  relative  to 
an  axis  perpendicular  to  the  direction  in 
which  the  maximum  deviation  is  sought ; 

and  y  =  the  distance  from  the  centre  of  gravity  of  the 
section  to  the  edge  furthest  from  the  centre 
of  pressure  measured  along  an  axis  passing 
through  the  centre  of  Dressure  and  the  centre 
of  gravity. 
Following  are  the  more  common  sections  of  foundations 

with  the  corresponding  values  of  x0: 

Rectangle .  .J.  =  bh,  x0  =  J5; 

Circle A  =  itr*,  x  =    d 


68  FOUNDATIONS   UPON  EARTH. 

Hollow  rectangle: 


Hoi.  squared  =  V  -  A",       *0  =  J  (l  +  *£)  ; 


Hoi.  circle.^  =  n(f  -  r"),  ^0  =      l  + 

The  ultimate  compressive  strengths  of  various  rocks  used 
in  foundations  are  approximately,  for 

Granite  ..............  ____  12800  pounds  per  square  inch. 

Sandstone  ................  9800  "  "  "  " 

Soft  sandstone  ...........  3000  "  "  "  " 

Strong  limestone  .........  8500  "  "  "  " 

Weak  limestone  ...........  3000  "  «  «  « 

Hard  red  brick  ...........  3000  "  "  « 

Common  brick  .......  ....  1000  «  "  «  " 

Portland  cement  concrete: 

1  month  old  ...........  1000  "  «  «  " 

12  months  "  ...........  6000  "  «  «  <tf 

Rosendale  cement  concrete  : 

6  months  old  ...........  1200  «  «  «  « 

FOUNDATIONS  UPON  EAKTH. 

Firm  Earth.  —  Earth  which  has  an  angle  of  repose  of  at 
least  27°  may  be  considered  as  firm,  and  for  foundation 
purposes  requires  little  preparation  other  than  the  excava- 
tion of  a  trench  or  pit,  and  making  the  surface  receiving 
the  masonry  level.  From  Table  II  it  is  seen  that  sand, 
gravel,  and  damp  clay  are  classed  as  firm  soils;  however, 


FOUNDATIONS   UPON  EARTH.  69 

these  may  become  so  saturated  with  water  that  their  angles 
of  repose  will  become  considerably  less  than  27°,  hence  pre- 
cautions must  be  taken  against  too  much  water  by  draining 
the  ground  in  the  immediate  vicinity  of  the  foundation. 
Particular  care  must  be  taken  in  the  case  of  clay,  or  sand 
which  will  become  a  kind  of  quicksand  when  saturated  with 
water. 

Before  attempting  to  design  a  foundation,  the  character 
of  the  earth  must  be  determined  either  by  test  excavations, 
borings,  or  from  the  experience  of  others.  It  often  happens 
that  from  all  surface  indications  the  earth  appears  to  be 
firm,  but  upon  excavating  it  is  found  there  is  a  stratum  of 
semi-fluid  mud  or  quicksand  underneath;  in  such  cases 
care  must  be  taken  to  determine  the  minimum  thickness  of 
the  stratum  of  firm  earth,  for  if  too  thin  it  will  not  be  safe 
to  build  upon,  and  then  a  foundation  has  to  be  prepared 
according  to  some  of  the  methods  described  later. 

Considering  the  earth  as  a  homogeneous  granular  mass, 
the  supporting  power  at  any  depth  can  be  computed  when 
the  angle  of  repose  0  is  known.  Some  practical  men  object 
to  any  theoretical  formulas  being  employed  in  connection 
with  the  determination  of  the  bearing  or  supporting  power 
of  earth,  claiming  that  the  assumptions  upon  which  the 
formulas  are  based  are  rarely  if  ever  found  in  practice. 
This  is  probably  true  to  a  certain  extent,  yet  the  theoretical 
formulas  are  upon  the  safe  side,  and  do  not  lead  to  absurd 
results;  in  fact,  the  results  obtained  by  their  judicious 
application  agree  very  well  with  the  practice  of  the  best 
engineers. 

If  p  —  the  maximum  supporting  power  per  square  foot 

of  earth ; 

y  =  the  weight  of  one  cubic  foot  of  earth; 
0  =  the  angle  of  repose; 


TO  FOUNDATIONS   UPON  EARTH. 

and  x'  =  the  depth  of  the  plane  below  the  surface  upon 
which  the  maximum  supporting  power  is 
desired  ; 

then 


And  if  p'  is  the  minimum  intensity  of  the  pressure  upon  the 
earth  which  is  allowable  for  the  stability  of  the  earth  and 
the  foundation  with  its  load, 

/  =  X"y  {  J~  ™-£  }  '  (see  page  40),   .    .     (2) 

where  x"  is  the  depth  of  the  plane  considered  below  the 
surface  of  the  earth. 

The  above  equations  neglect  any  friction  between  the 
earth  and  the  masonry  of  the  foundation.  In  deep  foun- 
dations this  is  a  large  factor  on  the  safe  side. 

If  the  surface  of  the  earth  is  level,  then  x'  —  x"  ;  and 
further,  if  the  earth  is  subjected  to  a  uniformly  distributed 
load  only  the  average  intensity  need  be  considered. 

Equation  (2)  is  considerably  different  from  that  given  by 
Eankine,  and  writers  who  have  followed  him,  in  this,  that 
they  consider  the  minimum  intensity  allowable  to  be  equal 
to  x"y  =  the  average  intensity  of  the  pressure  upon  a  plane 
at  a  depth  x"  in  an  unlimited  mass.  This  does  not  appear 
to  the  writer  to  be  a  logical  treatment  of  the  subject,  if  the 
mass  has  an  angle  of  repose  greater  than  zero,  and  the  maxi- 
mum intensity  allowable  be  determined  as  a  function  of 
this  angle. 

According  to  the  assumption  of  Rankine,  it  would  appear 
that  if  a  box  without  a  bottom  were  sunk  into  a  mass  of 
perfectly  dry  sand  it  would  be  filled  from  the  bottom  until 


FOUNDATIONS   UPON  EARTH.  71 

the  surfaces  without  and  within  were  at  the  same  level ;  but 
this  does  not  take  place,  and  would  not  even  if  the  sides  of 
the  box  were  frictionless.  The  sand  only  fills  the  box  par- 
tially, or  until  the  requirements  of  equation  (2)  are  fulfilled. 
Hence  it  seems  to  the  writer  that  if  the  maximum  intensity 
is  a  function  of  0,  the  value  of  the  minimum  intensity 
must  be  also. 

From  equations  (1)  and  (2)  it  is  evident  that  the  allow- 
able intensity  upon  the  earth  of  any  pressure  or  load  com- 
monly called  the  supporting  power  varies  directly  as  the 
depth,  as  long  as  0  remains  unchanged ;  hence  all  tables  of 
supporting  powers  of  earth  are  of  little  value  unless  the 
depth  of  the  foundation  upon  which  they  are  based  is 
known.  Unfortunately  this  is  omitted  in  most  cases,  and 
only  the  character  of  the  earth  is  given.  The  depth  to 
which  foundations  must  be  sunk  in  many  localities  has  a 
minimum  value  governed  by  the  depth  to  which  frost  ex- 
tends. This  is  not  always  true,  however,  as  in  Terre  Haute, 
Indiana,  frame  houses  and  brick  blocks  two  and  one-half 
stories  high  are  constructed  practically  upon  the  surface, 
the  sod  only  being  removed.  The  width  of  the  foundation 
is  not  excessive,  but  on  the  contrary  narrow.  No  serious 
settlement  results,  owing  to  the  character  of  the  earth, 
which  is  very  sandy,  and  will  not  retain  sufficient  moisture 
to  permit  frost  action  to  heave  the  structures.  The  actual 
load  per  square  foot  supported  by  the  soil  is  about  one  ton. 
If  x'  be  taken  as  one  foot,  y  as  100  pounds,  and  p  as  2000 
pounds,  then  from  equation  (1)  0  is  about  39°,  which  is 
below  the  actual  value. 

The  above  case,  however,  may  be  called  an  exception  to 
the  general  rule  that  all  foundations  must  be  sunk  below 
the  action  of  frost,  or  to  a  depth  of  three  feet  or  more 
according  to  the  locality. 


72  FOUNDATIONS   UPON  EARTH. 

For  convenience  the  values  of 


sn 


and 


are  given  in  the  following  table  : 


-  sn 


d) 

/l+siri<f>\2 

/I  -  sin  <»y 

fL 

/l+sin<j,y 

/I  -sin<£y 

9 

\1  -  sin  <£/ 

M  -f-  sin  <£/ 

9 

M  _  sin  <£/ 

Vl-fsin<£/ 

0 

.00 

1.00 

23 

5.21 

0.19 

5 

.42 

0.70 

24 

5.62 

0.18 

6 

.52 

0.66 

25 

6.07 

0.16 

7 

.63 

0.61 

26 

6.56 

0.15 

8 

.75 

0.57 

27 

7.09 

0.14 

9 

.88 

0.53 

28 

7.67 

0.13 

10 

2.02 

0.50 

29 

8.30 

0.12 

11 

2.16 

0.46 

30 

9.00 

0.11 

12 

2.32 

0.43 

31 

9.76 

0.10 

13 

2.50 

0.40 

32 

10.59 

0.09 

14 

2.68 

0.37 

33 

11.50 

0.09 

15 

2.88 

0.35 

34 

12.51 

0.08 

16 

3.10 

0.32 

35 

13.62 

0.07 

17 

3.33 

0.30 

36 

14.84 

0.07 

18 

3.59 

0.28 

37 

16.18 

0.06 

19 

3.86 

0.26 

38 

17.67 

0.06 

20 

4.22 

0.24 

39 

19.64 

0.05 

21 

4.48 

0.22 

40 

21.16 

0.05 

22 

4.83 

0.21 

Having  determined  upon  the  depth  to  which  it  is  ex- 
pedient to  extend  the  foundation,  a  minimum  value  of  0 
must  be  assumed  from  a  knowledge  of  the  earth,  and  then 
the  allowable  bearing  or  supporting  power  can  be  found 
from  equations  (1)  and  (2) ;  or  if  the  supporting  power  is 
assumed,  the  minimum  depth  to  which  the  foundation  must 
be  sunk  can  be  found  from  the  same  equations. 

The  proper  proportions  of  the  foundation  are  most  easily 
obtained  from  the  following  equations,  which  are  deduced 


FOUNDATIONS    UPON  EARTH. 


73 


for  a  few  of  the  ordinary  forms  and  conditions.  All 
masonry  foundations  are  usually  trapezoidal  in  section,  and 
hence  formulas  based  upon  this  form  can  be  applied  to 
stepped  foundations  without  serious  error. 

CASE  I.  Given  a  uniformly  distributed  load  to  be  sup- 
ported by  symmetrical  trapezoidal  foundation  sunk  to  a 
known  depth,  to  determine  the  minimum  width  of  the  base 
of  the  foundation. 


l«—     -     -     -  B"  A 

FIG.  35. 
Section  of  Wall  and  Foundation. 

Let  G  =  the  total  weight  to  be  supported  less  that  of  the 

foundation ; 

G'  =  G  +  weight  of  the  foundation ; 
and  B"  =  minimum  breadth  of  the  foundation. 
Assuming  xf ',  the  value  of  p  is 


From  the  figure 


7?  J_  7?" 
G'  =  a  +  W  -        —x'  = 


74  FOUNDATIONS   UPON  EARTH. 

or 

B"  = 


2p  -  Wx' 

The  above  formula  applies  to  a  wall  one  foot  long. — In 
case  of  an  isolated  pier,  the  value  of  x'  can  be  found  as 
above.  B"  may  be  assumed  and  a  rough  calculation  made 
to  determine  if  the  average  pressure  upon  the  earth  is  equal 
to  or  less  than  p.  A  second  trial  usually  determines  the 
proper  value  for  B".  The  exact  formula  for  the  determi- 
nation of  the  dimensions  of  a  square  or  rectangular  foun- 
dation with  stepped  sides  is  an  equation  of  the  second 
degree. 

Ex.  1.  A  trapezoidal  foundation  5  feet  broad  on  top 
has  to  support  50,000  pounds  per  lineal  foot  in  length,  in 
earth  having  a  minimum  angle  of  repose  of  30°.  The 
maximum  depth  to  which  the  foundation  is  to  be  sunk  is 
5  feet;  determine  B"  and  p,  when  y  =  100  pounds  and 
W  =  150  pounds. 

From  (1) 

p  —  5-100-9  =  4500  pounds— say  4000; 
then 

_  100000  +  3750 

8000  -  750 

or  the  proper  width  of  the  base  is  about  14.5  feet. 

Ex.  2.  A  cast-iron  plate,  2  feet  square  under  a  column, 
transmits  a  load  of  20,000  pounds  to  a  masonry  foundation 
3  feet  square.  How  deep  must  this  be  sunk  in  earth  when 
0  =  30°,  y  =  100  pounds,  and  W=  150  pounds? 

Neglecting  the  weight  of  the  masonry  in  the  foundation, 
the  intensity  of  the  pressure  upon  the  earth  is  about  2200 
pounds;  then  from  (1)  x'  =  about  2.5  feet — say  3  feet. 


FOUNDATIONS   UPON  EARTH. 


75 


The  actual  intensity  of  the  pressure  upon  the  earth  is 

now  — - —  -   =   2670   pounds.      Substituting  this 

y 

value  of  p  in  (1)  and  solving  for  a/,  its  value  is  2.96  feet; 
hence  3  feet  is  the  required  depth  of  the  foundation. 

The  weight  of  the  earth  supported  by  the  masonry  of  the 
foundation  is  neglected. 

CASE  II.  U asymmetrical  distribution  of  pressure  upon 
the  base  of  a  foundation. 


FIG.  36. 
Section  of  Wall  and  Foundation. 


One  of  the  many  examples  of  pressure  unevenly  dis- 
tributed upon  the  bed  of  a  foundation  is  the  case  of  an 
outside  wall  of  a  building  located  very  near  the  property 
line  and  circumstances  prevent  encroaching  upon  the  neigh- 


76  FOUNDATIONS   UPON  EARTH. 

boring  property  to  any  great  extent.  Here  two  conditions 
must  be  fulfilled.  The  maximum  intensity  of  the  pressure 
p,  Fig.  36,  must  not  be  greater  than  the  supporting  power 
of  the  earth  at  the  depth  a/,  and  the  minimum  intensity  j/ 
must  not  be  so  small  that  the  earth  having  a  depth  x"  may 
tend  to  heave  the  foundation. 

Let  p0  =  the  average  intensity  of  the  pressure  upon  the 
base.     Then 


But 


Therefore 

B"  =  -x 


in  which  x"  is  determined  from  the  equation 

„     (  1  -  sin  0  )  f 
p'  =  x"y  \  — — : — ^  f  • 
r  (  I  +  sm  0  ) 

It  is  thus  possible  to  determine  B"  quite  easily,  but  the 
value  of  the  offset  z  so  that  p  and  p'  shall  have  their  proper 
values  must  be  either  found  by  trial  or  computation.  Since 
one  or  two  trials  are  sufficient  to  determine  z,  the  formula 
will  not  be  given  here. 

Ex.  3.  In  Fig.  36,  pa^e  75,  let  G  =  40,000  pounds, 
B  =  4  feet,  d  =  2  feet,  x'  '=  24  feet,  and  x"  =  4  feet.  If 
the  thrust  of  the  earth  be  neglected,  what  must  be  the 
width  of  the  base  of  the  foundation,  so  that  the  average 
pressure  per  unit  area  shall  not  exceed  4800  pounds,  and 
the  maximum  7000  pounds,  when  y  =  100,  W  =  150, 


FOUNDATIONS   UPON  EARTH.  77 

0  =  30°  ?     The  bulk  of  the  foundation  to  be  on  the  right 
of  the  centre  of  the  wall. 

First  determine  the  allowable  intensities, 

max  p   —  «V(9)        =  2400  X  9       =  21600  pounds. 
min       =  ^V(O.ll)  =  2400  X  0.11  =      264 
maxy  =  z";/(9)      =400X9       =    3600 
min       =  z'O.ll   =    400  X  0.11  =        44         « 


=    9'15 


From  the  formula  on  page  76  * 

ZG  +  BWx"      82400 


•    2Po  -  Wv"    •     960 

Take  10  feet  as  the  value  of  B"  \  then  the  weight  of  the 
masonry  in  the  foundation  is  4200  pounds,  and 


By  graphics  or  by  moments,  assuming  z  =  2  feet,  the  re- 
sultant pressure  cuts  the  base  0.94  foot  from  the  centre,  and 
hence  p  =  6900  pounds  and  p'  =  1940  pounds. 

The  above  width  of  base  and  the  intensities  just  ob- 
tained satisfy  all  the  conditions  of  the  problem,  though  the 
value  of  z  could  be  decreased  a  little,  increasing  the  in- 
tensity at  the  toe  and  decreasing  that  at  the  heel. 

Projection  of  Footing-courses.  —  Where  masonry  founda- 
tions are  stepped  as  is  the  usual  custom,  the  proper  offset 
for  each  course  may  be  determined  as  follows,  by  consider- 
ing each  offset  as  a  cantilevered  beam  of  stone  uniformly 
loaded: 

Let  o    =  the  offset  of  any  particular  course; 

p0  =  the  intensity  of  the  pressure  upon  the  base  of 
the  course; 


78  FOUNDATIONS   UPON  EARTH. 

t    =  the  thickness  of  the  course; 
R  =  the  modulus  of  rupture  of  the  material;  and 
F  =  the  factor  of  safety. 
Then 


or 


R   1 

0  =  t  \  I  ~c?  ;r~ 


In  case  the  intensity  of  the  pressure  is  not  uniform,  but 
varies  uniformly  from  one  side  to  the  other,  the  quantity 
pt  may  be  replaced  by  p,  the  maximum  intensity  for  the 
offset  on  the  side  having  the  greater  pressure,  and  by  pf,  the 
minimum  intensity  for  the  steps  or  offsets  on  the  side  of 
the  lesser  pressure  :  in  the  first  case  the  factor  of  safety 
will  be  slightly  increased  and  in  the  second  decreased. 

The  above  formula  is  applicable  only  when  the  stones 
project  less  than  half  their  length  and  when  thoroughly 
well  laid  in  cement  mortar. 

The  table  on  the  following  page  is  given  by  Prof.  Baker. 

Other  factors  remaining  the  same,  the  offsets  vary 
directly  as  the  square  roots  of  the  moduli  of  rupture  and 
inversely  as  the  factors  of  safety,  so  that  the  above  table 
can  be  applied  for  any  values  of  R  and  F  by  simple  pro- 
portion. 

Foundations  upon  80 ft  Earth. — When  a  foundation 
must  be  placed  upon  soft  earth  which  offers  no  particular 
difficulties  other  than  the  requirement  of  broadness  or 
depth  of  the  excavation,  considerable  expense  can  be 
avoided  by  excavating  the  soft  material  and  replacing  it 
by  firm  material,  or  by  driving  short  piles  spaced  about 


FOUNDATIONS   UPON  EARTH. 


T9 


three  feet  on  centres,  commencing  at  the  outer  limits  of 
the  foundation  and  working  towards  the  centre,  and  thus 
compressing  the  earth;  sometimes  holes  are  bored  and  filled 
with  sand,  making  sand-piles,  etc.  The  proper  depth  and 
spread  of  such  foundations  can  be  found  from  formulas 
(1)  and  (2)  by  including  the  prepared  earth  as  a  portion  of 
the  foundation. 

SAFE   OFFSETS   FOR   MASONRY  FOOTING  COURSES, 

IN   TERMS  OF   THE   THICKNESS   OF   THE  COURSE,  USING   10  AS  A   FAC- 
TOR  OF   SAFETY. 


Offsets   for  a  Pressure,  in 

Kind  of  Stone. 

Rin 

Tons  per  Sq.  Ft.,  on  the 
Bottom  of  the  Course  of 

Sq.  In. 

Masonry. 

0.5 

1.0 

2.0 

Bluestone  nagging  

2700 

3  6 

2  6 

1  8 

1800 

2  9 

2.1 

1  5 

Limestone  .  • 

1500 

2  7 

1  9 

1  3 

Sandstone  

1200 

2  6 

1  8 

1  3 

Slate  

5400 

5  0 

3  6 

2  5 

Best  hard  brick  

1500 

2.7 

1.9 

1.3 

Hard  brick  

800 

1  9 

1  4 

0  8 

(1  Portland) 

Concrete  \  2  sand         J-  10  days  old  . 

150 

0.8 

0.6 

0.4 

(3  pebbles    ) 

(  1  Rosendale  ) 
Concrete  •{  2  sand            [•  10  days  old 

80 

0.6 

0.4 

0.3 

(  3  pebbles       ) 

In  case  the  earth  has  sufficient  water  to  keep  the  founda- 
tion damp,  a  very  excellent  foundation  upon  soft  earth  is 
a  platform  of  timber  composed  of  heavy  sticks  laid  close 
together  in  layers,  every  alternate  layer  being  right-angled 
with  that  adjacent,  and  thoroughly  driftbolted  together. 
Another  method  is  to  form  a  grillage  of  the  timbers  and 
fill  the  spaces  around  the  sticks  with  concrete. 


80  FOUNDATIONS   UPON  EARTH. 

In  dry  soft  earth  the  timber  platform  may  be  replaced  by 
a  bed  of  concrete,  which  is  more  durable,  but  not  as  elastic. 
Recently  the  combination  of  iron  or  steel  beams  with 
concrete  has  been  successfully  employed  for  foundations 
upon  soft  earth  in  Chicago. 

The  safe  projection  of  the  timber  platform  or  one  of 
concrete  beyond  the  masonry  can  be  found  by  the  formula 
already  given. 

The  safe  projection  of  iron  or  steel  beams  can  be  found 
as  follows: 

Let  /  =  the  moment  of  inertia  of  the  section; 
li  =  the  depth  of  the  beam ; 
po  =  the  intensity  of  the  pressure  upon  the  bed  of 

the  foundation  transmitted  to  the  beam; 
R  =  the  modulus  of  rupture  of  the  material  com- 
posing the  beam; 
and      F  =  the  factor  of  safety. 
Then 


o    /»    i 

=2v^ 


In  case  the  pressure  upon  the  base  of  the  foundation  is 
not  uniform,  the  method  outlined  for  masonry  offsets  can 
be  applied  in  proportioning  the  offsets  of  steel  or  iron 
beams. 

The  following  table  of  the  safe  projections  of  steel  I 
beams  is  given  in  Carnegie's  Pocket  Companion.  .  .  > 


FOUNDATIONS   UPON  EARTH. 


81 


TABLE 

GIVING  SAFE  LENGTHS  OP  PROJECTIONS  "  O  "  IN  FEET  (SEE 
ILLUSTRATION),  FOR  "«"  =  1  FOOT  AND  VALUES  OF 

"     O"  RANGING  FROM    1    TO   5   TONS. 


Depth 
of 
Beam, 
in. 

Weight 
per 
Foot, 
Ibs. 

6  (Tons  per  Square  Foot.) 

1 

H 

u 

2 

at 

21 

3 

3* 

4 

4i 

6 

20 

80 

14.0 

12.5 

11.5 

10.0 

9.0 

9.0 

8.0 

7.5 

7.0 

6.5 

6.0 

20 

64 

12.5 

11.0 

10.0 

8.5 

8.0 

8.0 

7.0 

6.5 

6.0 

6-0 

5.5 

15 

80 

12.0 

10.5 

9.5 

8.5 

8.0 

7.5 

7.0 

6.5 

6.0 

5.5 

5.0 

15 

60 

10.5 

9.5 

8.5 

7.5 

7.0 

6.5 

6.0 

5.5 

5.5 

5.0 

5.0 

15 

50 

9.5 

8.5 

8.0 

7.0 

6.5 

6.0 

5.5 

5.0 

5.0 

4.5 

4.5 

15 

41 

8.5 

8.0 

7.0 

6.0 

6.0 

5.5 

5.0 

4.5 

4.5 

4.0 

4.0 

12 

40 

8.0 

7.0 

6.5 

5.5 

5.5 

5.0 

4.5 

4.0 

4.0 

3.5 

3.5 

12 

32 

7.0 

6.5 

5.5 

5.0 

4.5 

4.5 

4.0 

4.0 

3.5 

3.5 

3.0 

10 

33 

6.5 

6.0 

5.5 

4.5 

4.5 

4.0 

4.0 

3.5 

3.5 

3.0 

3  0 

10 

25.5 

5.5 

5.0 

4.5 

4.0 

4.0 

3.5 

3.5 

3.0 

3.0 

2.5 

2.5 

9 

27 

5.5 

5.0 

4.5 

4.0 

4.0 

3.5 

3.5 

3.0 

3.0 

2.5 

2.5 

9 

21 

5.0 

4.5 

4.0 

3.5 

3.5 

3.0 

3.0 

2.5 

2.5 

2.5 

2.0 

8 

22 

5.0 

4.5 

4.0 

3.5 

3.5 

3.0 

3.0 

2.5 

2.5 

2  5 

2.0 

8 

18 

4.5 

4.0 

3.5 

3.0 

3.0 

3.0 

2.5 

2.5 

2.0 

2.0 

2.0 

7 

20 

4.5 

4.0 

3.5 

3.0 

3.0 

3.0 

2.5 

2.5 

2.0 

2.0 

2.0 

7 

15.5 

4.0 

3.5 

3.0 

2.5 

2.5 

2.5 

2.0 

2.0 

2.0 

2.0!  1.5 

6 

16 

3.5 

3.0 

3.0 

2.5 

2.5 

2.0 

2.0 

2.0 

1.5 

1.5 

1.5 

6 

13 

3.0 

3.0 

2.5 

2.5 

2.0 

2.0 

2.0 

1.5 

1.5 

1.5 

1.5 

5 

13 

3.0 

2.5 

2.5 

2.0 

2.0 

2.0 

1.5 

1.5 

1.5 

1.5 

1.5 

5 

10 

2.5 

2.5 

2.0 

2.0 

1.5 

1.5 

1.5 

1.5 

1.5 

4 

10 

2.5 

2.0 

2.0 

1.5 

1.5 

1.5 

1  5 

4 

7.5 

2.0 

2.0 

1.5 

1.5 

1.5 

1  5 

Above  table  applies  to  steel  beams.  Values  given  based  on  ex- 
treme fibre  stresses  of  16,000  pounds  per  square  inch. 

Pile  Foundation. — Pile  foundations  are  employed  in  all 
kinds  of  earth,  sometimes  to  save  expense  and  sometimes 
because  nothing  else  appears  to  be  as  good.  In  localities 
where  the  earth  is  uncertain  in  its  character  the  use  of 


82 


FOUNDATIONS  UPON  EAUTII. 


piles  enables  the  engineer  to  put  in  a  foundation  which  he 
foels  sure  is  safe,  as  a  single  pile  thirty  feet  long  will  sup- 
port several  tons  even  when  driven  into  mud,  the  load  in 
this  case  being  carried  almost  entirely  by  the  friction  of  the 


BASEMENT 

— 

in 

~ 

3 

vtiwsmxmsnm 

s 

'3//=fff 

\ 

+  0     J         MASONRY 

MASONRY       | 

I  BEAMS  IN  CONCRETE 

CONCRETE]      ^\ 

FIG.  37. 

mud  upon  the  surface  of  the  pile.  If  the  pile  is  driven 
through  the  mud  to  a  solid  stratum  below,  then  the  pile 
acts  as  a  column  more  or  less  supported  its  entire  length, 
and  consequently  able  to  carry  a  very  great  load. 

Piles  are  usually  spaced  about  three  feet  on  centres,  and 
the  tops  firmly  bedded  in  a  layer  of  concrete  or  stayed  by  a 
grillage  of  timber  or  by  a  combination  of  these  methods, 
the  object  being  to  thoroughly  and  evenly  distribute  the 
load  to  be  supported. 

The  supporting  power  of  a  pile  in  a  given  earth  can  be 
found  in  the  following  manner: 

Let  G'  =  the  total  load  to  be  supported  by  the  pile,  in- 
cluding the  weight  of  the  pile; 
p0  =  the  intensity  of  the  pressure  upon  the  bottom 

of  the  pile; 

A  =  the  superficial  area  of  the  pile  in  contact  with 
the  earth; 


FOUNDATIONS  UPON  £AKTI1.  S3 

and      f  =  a  factor  depending  upon  the  friction  resistance 

of  a  unit  area  of  the  surface  of  the  pile. 
Then  for  a  pile  having  a  diameter  of  d 


But 

-f-  sin  0  *  2 

G' 


and 


sn 


—  sm  < 
For  practical  purposes  this  may  be  written 


jliBin|,' 

'   (  1  —  sin  0  ) 


For  convenience  this  may  be  further  simplified  for 
special  cases. 

The  following  values  of  f  have  recently  been  given  by 
W.  M.  Patton,  based  upon  his  own  and  the  experience  of 
others : 

In  very  soft  silt  or  liquid  mud,   /  =  150  pounds  per  sq.  ft. 
In  ordinary  clay  or  earth  (dry),  /  =  300       "        "     "     " 
"     «      "      (wet),  /=  150       "        "     "    « 
In  compact  hard  clay,  /  =  300       "        "     "     " 

In  sand,  or  sand  and  gravel,        /  =  500       "        "     "    " 


*  This  formula  was  suggested  by  reading  W.  M.  Pattern's  article  on 
piles  in  his  "  Practical  Foundations." 


84  FOUNDATIONS   UPON  KARTH. 

For  the  silt  of  swamps,  muds,  etc.,  <p  is  very  nearly  if 
not  quite  zero.  So  as  to  be  on  the  side  of  safety,  0  will  be 
taken  as  zero,/=  150  pounds.  Then 


, 


_ 

120  +  450  ~~  570  '  600* 


a  very  simple  formula. 

For  moist  clay,  0  =  about  17°,  y  =  120  pounds,  and 
f=  150  pounds.     Then 

G'  G' 

120  .  3i  +  450        850  * 

For   dry,  compact  sand,  0  =  27°,  y  =  106  pounds,  and 
f=  500  pounds.      Then 


9— 

~ 


. 

107.7  +  1500  ~~  2249'      2300* 


In  a  similar  manner  the  safe  load  for  a  pile  in  any  earth 
can  be  determined  when  0  and  /  are  kno\^  n.  These  quan- 
tities must  be  the  result  of  experiment.  Any  formula  which 
does  not  include  these  factors  is  incomplete,  and  neglects 
the  factors  upon  which  the  supporting  power  of  the  pile 
directly  depends. 

The  character  of  the  earth  through  which  the  pile  is  to 
be  driven  can  be  determined  by  borings,  and  thus  0  and  y 
determined  upon. 

The  value  of  /can  be  found  by  studying  the  behavior  of 
piles  already  driven  in  similar  earth.  Thus  it  appears  that 
the  above  formula  must  be  as  accurate  in  results  and  as 
safe  in  application  as  the  majority  of  the  formulas  used  by 
engineers  in  proportioning  structures. 


FOUNDATIONS   UPON  EARTH.  85 

The  formula  is  independent  of  the  means  by  which  the 
pile  is  driven,  as  ought  to  be  the  case,  since  very  of  ten  piles 
are  sunk  by  water-jets,  or  even  by  working  them  backwards 
and  forward,  making  the  formulas  depending  upon  the 
weight  of  a  driving-hammer,  its  fall,  and  the  penetration  of 
the  pile  during  the  last  few  blows  useless.  Two  of  the 
most  simple  of  the  many  formulas  of  this  class  are  those 
of  Trautwine  and  the  Engineering  News,  viz.  : 


(Eng.News); 


where       W'=  the  weight  of  hammer  in  tons; 
G'  —  the  safe  load  in  tons; 
W  =  the  weight  of  the  hammer  in  pounds; 
h  =  the  fall  of  the  hammer  in  feet; 
and  a  =  the  average  penetration  of  the  pile  in  inches 

during  the  last  few  blows. 

Screiv-pile.  —  Screw-piles  are  usually  round,  and  have  at 
the  bottom  a  cast  or  wrought  iron  screw.  The  piles  are  of 
wood,  cast  iron,  or  wrought  iron.  The  diameter  of  the 
screw  is  from  two  to  eight  times  the  diameter  of  the  pile, 
and  its  pitch  from  one  fourth  to  one  half  its  diameter. 
The  screw  seldom  has  but  one  turn.  The  piles  are  sunk 
by  turning  them  by  means  of  levers  or  by  power.  (Fig. 
45.) 

The  load  which  the  pile  will  carry  depends  principally 
upon  the  supporting  power  of  the  earth  at  the  depth  of 
the  screw  and  the  area  of  the  screw,  though  in  all  cases 
there  is  more  or  less  frictional  resistance  upon  the  surface  of 
•the  pile  proper,  If  x'  is  the  depth  of  the  screw  and  pu  the 


86  FOUNDATIONS   UPON  EARTH. 

allowable  intensity  of  the  pressure  upon  the  earth  at  that 
depth,  then 


_  x'  j  1  -  sin  0  I a 
Po  ~  Y  (  1  +  sin  0  [  ' 


sin  0 

Screw-piles  can  be  advantageously  employed  for  support- 
ing structures  above  water  where  the  upper  ends  of  the 
piles  can  be  used  as  columns.  They  are  chiefly  employed 
in  light-house  construction. 

Sheet-piles. — Sheet-piles  are  usually  of  wood  in  the  form 
of  planks,  and  are  driven  as  closely  together,  edge  to  edge,  as 
possible,  the  object  being  to  form  a  water-tight  barrier. 

To  make  the  joints  tight  the  planks  are  of  tened  tongued 
and  grooved.  A  patent  sheet-pile  is  formed  by  bolting 
together  three  planks  of  equal  width,  so  that  the  middle 
plank  will  form  the  tongue  on  one  side  and  the  outside 
planks  the  groove  on  the  other  side.  Sheet-piles  are  also 
employed  to  confine  soft  earths. 


FOUNDATIONS  UNDER  WATER  AND 
DEEP  FOUNDATIONS. 

Foundations  under  water  differ  in  general  but  little  from 
those  upon  dry  earth,  the  effect  of  water,  ice,  etc.,  upon 
the  structure,  however,  constitute  additional  problems  to 
be  solved  for  each  locality. 

A  few  of  the  various  methods  employed  in  placing  foun- 
dations under  water  or  at  great  depths  will  be  very  briefly 
described. 

Coffer-dams. — A  coffer-dam  is  merely  a  tight  wall  sur- 
rounding the  locality  where  the  foundation  is  to  be  placed, 
excluding  water  from  the  enclosure,  which  can  be  pumped 
dry  and  the  surface  prepared  to  receive  the  foundation. 

In  quiet  and  shallow  water  the  dam  may  be  made  of 
earth,  or  sheet-piles  banked  with  earth. 

In  deep  water  large  piles  are  driven  every  few  feet  in 
two  rows  around  the  site,  to  which  horizontal  timbers  are 
bolted,  acting  as  guides  and  supports  to  a  double  row  of 
sheet-piles,  between  which  is  placed  puddled  earth.  To 
prevent  bending,  the  large  piles  are  cross-tied  with  bolts. 

The  space  enclosed  should  be  somewhat  larger  than 
required  by  the  foundation,  to  allow  room  for  materials, 
etc.  (Fig.  46.) 

Timber  Cribs. — A  timber  crib  is  a  box  built  of  large 
timbers  and  divided  into  cells  by  cross  partitions.  The 
joints  and  splices  of  the  timbers  employed  are  arranged  so 
that  walls  and  partitions  are  thoroughly  tied  together.  In 

87 


88  FOUNDATIONS   UNDER   WATER. 

case  a  tight  wall-crib  is  wanted  the  timbers  may  be  dapped 
one  fourth  their  depth  on  both  sides  or  halved  together. 
Cribs  are  built  in  the  shape  best  suited  to  the  purpose  for 
which  they  are  to  be  used.  They  are  usually  constructed 
at  some  convenient  point  near  the  site  of  the  foundation, 
and  then  towed  to  the  place  where  they  are  to  be  sunk. 
In  constructing  the  crib  a  few  of  the  cells  are  planked  near 
the  bottom.  These  are  filled  with  stone  until  the  crib 
sinks  to  the  surface  previously  prepared  to  receive  it.  The 
other  cells  are  now  filled  with  stone  and  the  regular  mas- 
onry commenced.  Sometimes  the  top  of  the  crib  is  planked 
over  before  the  masonry  is  started.  (Fig.  44.) 

The  surface  which  is  to  receive  the  crib  may  be  soft 
mud,  riprap,  rock,  or  piles.  The  crib  is  allowed  to  sink  into 
the  mud  and  to  rest  upon  riprap  which  has  been  levelled. 
If  the  surface  is  level  rock,  the  crib  is  merely  sunk;  but  if 
the  rock  is  uneven,  it  is  either  levelled  or  the  crib  is  sunk 
until  it  just  touches  rock  at  some  point,  when  riprap  is 
thrown  around  and  under  the  crib. 

Timber  cribs  are  extensively  employed  in  various  classes 
of  engineering  works  for  both  temporary  and  permanent 
structures. 

In  permanent  structures  the  timbers  supporting  mas- 
onry, etc.,  should  always  be  under  water. 

Timber  cribs  are  sometimes  used  as  coffer-dams  by  mak- 
ing the  outside  cells  water-tight.  The  crib  is  sunk  into 
the  mud,  or  the  bottom  edges  banked  with  earth,  etc., 
until  the  interior  can  be  kept  dry  by  pumping. 

Open  Caissons. — An  open  caisson  is  a  strong  water-tight 
box  which  is  floated  to  the  site  of  the  foundation  and  sunk  to 
its  place  by  the  masonry  proper,  which  is  built  inside  the 
box.  After  the  bottom  has  reached  its  position  and  the 
top  of  the  masonry  is  above  water,  the  sides  are  removed, 


FOUNDATIONS  UNDER   WATER.  89 

leaving  the  bottom  of  the  box  as  a  platform  supporting  the 
masonry.  The  surface  to  receive  an  open  caisson  is  pre- 
pared by  dredging,  throwing  in  riprap,  driving  piles,  etc., 
as  best  suits  the  locality.  (Fig.  47.) 

Gushing  Cylinder  Piers. — A  cluster  of  piles  is  first 
driven  as  closely  together  as  possible,  and  their  tops 
thoroughly  bolted  one  to  the  other.  Then  an  iron  cylinder 
is  placed  around  the  cluster  and  built  up  in  sections 
until  the  top  is  above  water.  Then  the  cylinder  is  made 
to  sink  by  dredging  out  the  material  inside  by  water- jets, 
by  disturbing  the  material  around  the  edges,  etc.,  until  a 
desired  depth  is  reached,  sections  being  bolted  to  the  top 
of  the  cylinder  as  needed.  The  cylinder  is  now  filled  with 
concrete  to  the  top  and  covered  with  an  iron  cap  which 
receives  the  load  to  be  carried.  The  size  and  number  of 
cylinders  employed  depends  upon  the  superstructure. 

For  ordinary  bridges  two  cylinders  cross-braced  form  a 
pier. 

The  supporting  power  depends  upon  the  piles  principally, 
though  the  friction  upon  the  outside  of  the  cylinders  offers 
some  resistance  to  settlement. 

Pneumatic  Caissons. — A  pneumatic  caisson  is  essentially 
an  air-tight  box  with  the  open  side  imbedded  in  earth,  from 
which  the  air  is  pumped  to  allow  the  box  to  sink  or  into 
which  air  is  pumped  to  prevent  sinking.  In  water  the 
caisson  usually  carries  a  water-tight  timber  crib,  which  in 
turn  supports  a  timber  coffer-dam,  the  crib  enabling  the 
structure  to  be  loaded  with  stone  according  to  the  require- 
ments of  the  sinking  operation,  and  the  coffer-dam  keeping 
the  water  out  near  the  surface.  Various  combinations  of 
caisson,  crib,  and  coffer-dam  are  made,  however,  to  suit 
conditions.  (Fig.  48.) 

The  ordinary  method  of  sinking  caissons  is  to  pump  in 


90 


FOUNDATIONS    UNDER    WATER. 


enough  air  to  exclude  water  from  the  chamber,  while 
laborers  dig  out  the  material  over  the  surface  and  near  the 
edges  of  the  chamber,  this  material  being  removed  by 
various  methods  such  as  pumps,  lifts,  etc.  When  sufficient 
material  has  been  removed,  all  the  laborers  leave  the 
caisson,  leaving  one  man  only  who  watches  for  leaks;  the 
air-pressure  is  then  lowered  a  little,  and  the  caisson  with 
its  superstructure  sinks.  This  process  is  repeated  until  a 
solid  foundation  is  reached,  when  the  caisson  is  filled  with 
concrete,  as  also  are  the  cribs,  etc.,  if  any,  above  the 
caisson. 


TYPES  OF  EXISTING  FOUNDATIONS. 


JTP\ 


FIG.  38. 

Concrete  Pier  used  as  Foundation  for  Elevated  Railroad  Columns 
(Engineering  and  Building  Record,  Sept.  14,  1895.) 


TYPES  OF  EXISTING  FOUNDATIONS, 


ROCK 
FIG.  39. 

Elevation  of  Masonry  Pier  with  Bottom  Course  of  Concrete.  Illus- 
trating the  removal  of  rotten  rock  and  the  levelling  of  the  rock 
surface.  (Marent  Gulch  Viaduct,  N.  P.  R.  R.;  Trans.  Am.  Soc. 
G.  ^.,Sept.,  1891.) 


92 


TYPES  OF  EXISTING  FOUNDATIONS. 


"1 


FIG.  40. 

Flevation  of  another  Pier  of  the  Marent  Viaduct  Foundations.     Show- 
ing the  application  of  piles  and  concrete  to  obtain  a  solid  foundation. 


TYPES  OF  EXISTING  FOUNDATIONS. 


93 


FIG.  41. 

Elevation  of  a  Pier  in  the  Foundation  of  a  Chicago  Grain  Elevator. 
Illustrating  the  use  of  piles  and  a  wooden  platform  in  soft  ground. 
Piles  are  from  20  to  40  feet  long,  and  reach  hardpan.  Twelve 
piles  are  placed  under  each  post,  and  each  pile  supports  a  load  of 
about  22  tons.  (Engineering  and  Building  Record,  Nov.  12.  1895.) 


TYPES  OF  EXISTING   FOUNDATIONS. 


W/////^^^^^^^^^///////^ 

n  UCK 

FIG.  42. 

End  Elevation  of  Masonry  Pier  supporting  Stone  Arches  of  Wash- 
mo-ton  Bridge.  Illustrating  the  use  of  concrete  to  level  the  rock 
surface  to  receive  masonry. 


FIG.  43. 

Section  through  Centre  of  Foundation  of  Pivot  Pier  of  Grand  Forks 
Bridge.  Illustrating  the  use  of  piles,  wooden  platform,  and  rip- 
rap. (Baker.} 


TYPES  OF  EXISTING  FOUNDATIONS. 


95 


FIG.  44. 

End  Elevation  of  Foundation  of  Pier  of  Croix  River  Bridge, 
trating  the  use  of  timber  erib  and  piles.     (Baker.) 


Illus. 


]fr~ 

6*< 

K 

tu 

I- 

1 

i=~  ° 

OW  TO  WATER 

^   i!  v- 

?._  _J__" 

•^ 

„. 

FIG.  45. 

Mobile  River  Bridge  Piers.  Composed  of  two  rows  of  screw-pilr s, 
about  9  feet  centre  to  centre,  with  piles  spaced  about  8  feet  apart. 
(See  Engineering  News,  vol.  xiii.  p.  210.) 


TYPES  OF  EXISTING  FOUNDATIONS. 


[PLATFORM 

J^N 

\\\Nfc^siK\\vtS^; 

EM 
U 

•••m  ..  *_-'.^ 

'  —  ' 

LJ 

_J 

— 

Q 

;- 

Q 

^ 

- 

LU 

Q. 

I 

CO 

"% 

7Z5 

>t 

.  46. 


Sketch  showing  Cross-section  of  Coffer-dam, 


J L 


FIG.  47. 
Sketch  showing  Essential  Features  of  Open  Caisson. 


TYPMti  OF  EXISTING  FOUNDATIONS. 


97 


r. 


-•  -54-6 


12'x  12" 

\ 

12"x  12" 

I 

1 

1     K 

12x12 

1     '  i 

12"  x  12" 

1 

1= 

IF"                                           TARRED  PAPER 

n* 

BRACES  ABOUT  1C/APART            £_ 

f 

OAK 


Section  of  One  of  the  Caissons  employed  in  the  Foundations  of  the 
Piers  for  the  Washington  Bridge. 


REFERENCES. 

EAETH-PEBSSUBB   AND   EETAINING-WALLS. 


A  brief  outline  of  the  theories  advanced  by  the  follow- 
ing writers  can  be  found  in  "  Neiie  Theorie  des  Erd- 
druckes,"  Dr.  E.  Winkler,  Wien,  1872: 

D' Antony,  Hoffmann,  Poncelet, 

Ande,  Holzhey,  Prony, 

Andoy,  de  Lafont,  Eankine, 

Belidor,  Levi,  Rebhann, 

Blaveau,  de  Koszegh  Martony,  Rondelet, 

Bullet,  Maschek,  Saint-Guilhem, 

Considere,  Mayniel,  Saint- Venant, 

Coulomb,  Mohr,  Sallonnier, 

Couplet,  Montlong,  SchefHer, 

Culmann,  Moseley,  Trincaux, 

Fran^ais,  Navier,  Vauban, 

Gadroy,  Ortmann,  Winkler, 

Gauthey,  v.  Ott,  Woltmann. 

Hagen,  Persy, 

AUDE.  Poussee  des  Terres.     Nouvelles  experiences  sur  la 

poussee  des  terres.     Paris,  1849. 

BAKER-CURIE.  Note  sur  la  brochure  de  M.  B.  Baker  theorie. 
Annales  des  Ponts  et  Chaussees,  pp.  558-592,  1882. 

The  actual  lateral  pressure  of  earthwork.     Van  Nos- 

trand's   Magazine,  xxv,  1881;    also   Van   Nostrand's 
Science  Series,  No.  56. 

99 


100  REFERENCES. 

BOUSSIN^ESQ.  Complement  a  de  precedentes  notes  sur  la 
poussee  des  terres.  *Annales  P.  et  C.,  1884. 

BOUSIN.  Equilibrium  of  pulverulent  bodies.  The  equilib- 
rium of  earth  when  confined  by  a  wail.  fVan  N.,  188 1. 

CAIN.   Modification  of  Weyrauch's  Theory.    VanN.,  1880. 

-  Earth-pressure.    Modification  of  Weyrauch's  Theory. 
Criticism  of  Baker's  articles.     Van  N.,  1882. 

•—  Uniform  cross-section,  and  T  abutments :  their  proper 
proportions  and  sizes,  deduced  from  Rankine's  general 
formulas.  Van  N.,  1872. 

-  Practical    designing    of    retaining-walls.      Van    N. 
Science  Series,  No.  3,  1888. 

CHAPERON.  Observations  sur  le  rnemoire  de  M.  de  Sazilly 
(1851).  Stabilite  et  consolidation  des  talus.  Annales 
P.  et  C.,  1853. 

CONSIDERE.  Note  sur  la  poussee  des  terres.  Annales  P.  et 
C.,  1870. 

COUSINERY.  Determination  graph ique  de  Tepaisseur  des 
inurs  de  soutenement.  Annales  P.  et  C.,  1841. 

DE  LAFONT.  Sur  la  poussee  des  terres  et  sur  les  dimensions 
a  donner,  suivant  leurs  profils,  aux  murs  de  soutene- 
ment et  de  reservoirs  d'eau.  Annales  P.  et  C.,  1866. 

DE  SAZILLY.  Sur  les  conditions  d'equilibre  des  massifs  de 
terre,  et  sur  les  revetements  des  talus.  Annales  P.  et 
C.,  1851. 

EDDY.  Retaining-walls  treated  graphically.     VanN.,  1877. 

FLAMANT.  Note  sur  la  poussee  des  terres.  Annales  P.  et 
C.,  1882. 

Resume  d'articles  publics  par  la  Societe  des  Inge- 

nieures  Civils  de  Londres  sur  la  poussee  des  terres.  An- 
nales P.  et  C.,  1883. 

*  Annales  des  Ponts  et  Chaussees. 
f  Van  Nostrand's  Magazine, 


REFERENCES.  101 

FLAMANT.  Note  sur  la  poussee  des  terres.  Annales  P.  et 
C.,  1872. 

-  Memoire  sur  la  stabilite  de  la  terre  sans  cohesion  par 
W.   J.  Macquorm  Rankine    (Extrait    1856-57).      An- 
nales P.  et  C.,  1874. 

GOBIN.  Determination  precis  de  la  stabilite  des  murs  de 
soutenement  et  de  la  poussee  des  terres.  Annales  P. 
et  C.,  1883. 

GOULD.   Theory  of  J.  Dubosque.     Van  N.,  1883. 

-  Designing.     Van  N.,  1877. 

JACOB.  Practical  designing  of  retaining-walls.  Van  N., 
1873;  also  Van  N.  Science 'Series,  No.  3. 

JACQUIEE.  Note  sur  la  determination  graphique  de  la 
poussee  des  terres.  Annales  P.  et  C.,  1882. 

KLEITZ.  Determination  de  la  poussee  des  terres  et  eta- 
blissement  des  murs  de  soutenement.  Annales  P.  et 
C.,  1844. 

LAGKEUE.  Note  sur  la  poussee  des  terres  avec  ou  sans  sur- 
charges. Annales  P.  et  C..  1881. 

L'EVEILLE.  De  1'emploi  des  contre-forts.     Annales  P.  et  C. 

1844. 
LEYGUE.  Sur  les  grands  murs  de  soutenement  de  la  ligne 

de  Mezamet  a  Bedarieux.     Annales  P.  et  C.,  1887. 

-  Nouvelle   recherche  sur  la  poussee   des  terres  et  le 
profil  de  revetement  le  plus  economique.     Annales  P. 
et  C.,  1885. 

MEBEIMAN.  On  the  theories  of  the  lateral  pressure  of  sand 
against  retaining  walls.  (School  of  Mines  Quarterly.) 
Engineering  News,  1888. 

-  The  theory  and  calculation  of  earthwork.     Engineer- 
ing News,  1885. 

REBHANX.  Theorie  des  Erddruckes  und  der  Futtermauern. 
Wien,  1870  and  1871. 


102  REFERENCES. 

SAINT-GTJILHEM.  Sur  la  poussee  des  terres  avec  ou  sans 
surcharge.     Annales  P.  et  C.,  1858. 

ScHEFFLER-FouRNiE.  Traite  de  la  stabilite  des  construc- 
tions.    Paris,  1864. 

TATE.  Surcharged  and  different  forms  of  retaining  walls. 
Van  N.,  1873;  also  Van  N.  Science  Series,  No.  '; 
Also  published  by  E.  &  F.  N.  Spon. 

THORNTON.  Theory.     Van  N.,  1879. 

FOUNDATIONS. 

BAKER.  A  treatise  on  masonry  construction.     John  Wiley 

&  Sons,  N.  Y. 
PATTON.  A  practical  treatise  on  foundations.  John  Wiley 

&  Sons,  N.  Y. 
A  treatise  on  civil  engineering.  John  Wiley  &  Sons, 

N.  Y. 

For  articles  in  engineering  periodicals  the  reader  is  re- 
ferred to  a  "  Descriptive  Index  of  Current  Engineering 
Literature"  (1884-1891),  published  by  the  Board  of 
Managers  of  the  Association  of  Engineering  Societies. 
'•Engineering  Index/' Vols.  II  and  III,  published  by  the 
Engineering  Magazine. 


TABLES. 


Table  I  contains  the  crushing-strengths  and  the  average 
weights  of  stone  likely  to  be  used  in  the  construction  of 
retaining-  walls  and  foundations;  also  the  average  weights 
of  different  earths. 

Table  II  contains  the  coefficients  of  friction,  limiting 
angles  of  friction,  and  the  reciprocals  of  the  coefficients  of 
friction  for  various  substances. 

Tables  III,  I  V,  and  V  contain  the  values  of  the  coeffi- 
cients [see  equation  (!')]  (B),  (C),  (D)  and  (E),  where 


' 


t 

cos2  a:  cos  e'  (       cos  e 


cos  (e  —  a) 
and  (  EI  )  =  2  sin  a  sin  e  -  -  ---  -. 

cos  e 


The  tables  were  computed  with  a  Thacher  calculating  in- 
strument and  checked  by  means  of  diagrams.  It  is  believed 
that  they  are  correct  to  the  second  place  of  decimals;  an 
error  in  the  third  place  of  decimals  does  not  affect  the  re- 
sults for  practical  purposes. 

Table  VI  contains  the  natural  sines,  cosines  and  tan- 
gents. 

105 


106 


TABLES. 


TABLE  I. 

VALUES  OP    W 


Name  of  Substance. 

Crushing 
Lds.  in  tons 
per  sq.  ft. 

Average 
weight  in  Ibs. 
per  cu.  ft. 

Alabaster 

144 

Brick   best  pressed             

40  to  300 

150 

"       common  hard  

125 

soft  inferior  

100 

Chalk 

20  to  30 

150 

49.6  to  102 

Flint     

162 

Feldspar 

166 

Granite          . 

300  to  1200 

170 

Gneiss      ...                       ... 

168 

Greenstone  trap             .       

187 

Hornblende,  black  

203 

Limestones  and  Marbles,  ordinary  

250  to  1000 

J164.4 

Mortar  hardened  

(  168 
103 

Quartz    common  

165 

Sandstone 

150  to  550 

151 

Shales 

162 

Slate      .... 

400  to  800 

175 

Soapstone  

170 

VALUES   OP   y. 


Name  of  Substance. 


Earth    common  loam 

loose                     .  .           . 

72  to  80 

shaken     

82      9  'I 

«             <. 

rammed  moderately  

90      100 

Gravel   

90      106 

S;ind 

90      106 

Soft  flowing  mud.  .  .  . 

104      120 

Sand  perfectly  wet 

118      129 

Average 

weight  in  Ibs. 

per  cu.  ft. 


TABLES. 
TABLE  II. 

*  ANGLES  AND    COEFFICIENTS  OF  FRICTION. 


107 


tan  </>. 

«*> 

1 
tan  $ 

Dry  masonry  and  brickwork 
Masonry     and     brickwork 
with  damp  mortar       .... 

0.6to  0.7 
0.74 

31°  to  35° 
364° 

1.67  to  1.43 
1  35 

Timber  on  stone  
Iron  on  stone  
Timber  on  timber  
Timber  on  metals  

about  0.4 
0.7    to  0.3 
0.5     "0.2 
06"  0.2 

22° 
35°    to!6|° 

31°    "  1H° 

2.5 
1.43  to  3.  33 
2  "5 
1.67  "  5 

Metals  on  metals  
Masonry  on  dry  clay      .... 

0.25  "  0.15 
0  51 

14°     "    8|° 

27° 

4  "6.67 
1  96 

"         "    moist  clay  
Earth  on  earth 

0.33 
0  25  to  1  0 

18*° 
14°  to  45° 

3. 

4  to  1 

Earth   on    earth,  dry  sand, 
clay,  and  mixed  earth.  .  .  . 
Earth  on  earth,  damp  clay  . 
Earth  on  earth,  wet  clay. 
Earth  on  earth,  shingle  and 
gravel             ....         ... 

0.38  "0.75 
1.0 
0.31 

0  81 

21°  "  37° 
45° 

17° 

39°  to  48° 

2.63"  1.33 
3.23 
1  23  to  0  9 

*  From  Raukiue's  Applied  Mechanics. 


108 


TABLES. 


TABLE  III. 


€ 

a  =  5° 

a  =  6° 

a  =7° 

a  =  8° 

•a  =  y° 

(B) 

(B) 

(B) 

(B) 

(B) 

0 

1.004 

1.005 

1.007 

1.010 

1.012 

5 

1.012 

1.015 

1.018 

1.022 

1.026 

10 

1.019 

1.024 

1.029 

1.035 

1.040 

15 

1.027 

1.034 

1.041 

1.048 

1.055 

20 

1.036 

1.044 

1.052 

1.062 

1.071 

25 

1.045 

1.055 

1.065 

1.076 

1.088 

30 

1.055 

1.066 

1.079 

1.092 

1.105 

35 

1.065 

1.079 

1.094 

1.109 

1.124 

40 

1.078 

1.094 

1.111 

1.129 

1.147 

45 

1.093 

1.111 

1.181 

1.152 

1.173 

1           (C) 

(C) 

(C) 

(C) 

(C) 

\         0.008 

0.011 

0.015 

0.019 

0.024 

TABLE  IV. 


e 

a  =  5° 

a  =  6° 

a  =  7°                   a  =  8° 

a  =  9° 

(D) 

CD) 

(D) 

(D) 

(D) 

0 

0.992 

0.989 

0.985 

0.981 

0.976 

5 

1.008 

1.008 

1.006 

1.005 

1.003 

10 

1.023 

1.026 

1.028 

1.030 

1.031 

15 

1.040 

1.046 

1.051 

1.056 

1.060 

20 

1.057 

1.066 

1.075 

1.084 

1.092 

25 

1.0,5 

1  089 

1.102 

1.114 

1.125 

30 

1.096 

1.113 

1.180 

1.147 

1  .  163 

35 

1.118 

1    140 

1  .  1  64 

1.183 

1.204 

40 

1.144 

1.172 

1.199 

1  226 

1.253 

45 

1.174 

1.208 

1.242 

1.276       |       1.309 

TABLE  V. 


6 

a  =  5° 

a  =  6° 

a  =  7° 

a  =  8" 

a  =  9° 

(E) 

OB) 

(E) 

(E) 

(E) 

0 

0 

0 

0 

0 

0 

5 

0.015 

0.018 

0.021 

0.024 

0.027 

10 

0.031 

0.037 

0.043 

0.049 

0.055 

15 

0.046 

0.055 

0.065 

0.074 

0.083 

20 

0.061 

0.074 

0.086 

0.099 

0.112 

25 

0.076 

0.092 

0.108 

0.124 

0.140 

30 

0.091 

0.110 

0.130 

0.149 

0.169 

35 

0.106 

0.128 

0.151 

0.174 

0.197 

40 

0.120 

0.145 

0.172 

0.198 

0.225 

45 

0.134 

0.162 

0.192 

0.222 

0.25:3 

i 

TABLES. 


109 


TABLE  Ill—Continued. 


e 

<x=  10° 

a  =  11° 

a=  12° 

a  =  13° 

a  =.-  14° 

a?) 

(£) 

(#) 

(#) 

(#) 

0 

1.015 

1.019 

1.022 

1.026 

1.031 

5 

1.031 

1.037 

1.041 

1.047 

1.053 

10 

1.046 

.055 

1.061 

1.068 

1.076 

15 

1.063 

.073 

1.081 

1.090 

1.100 

20 

1.081 

.092 

1.103 

1.112 

1.125 

25 

1.099 

.112 

1.124 

1.136 

1.150 

30 

1.119 

.135 

1.151 

1.163 

1.179 

35 

1.141 

.159 

1.175 

1.195 

1.211 

40 

1.166 

.186 

1.205 

1.225 

1.245 

45 

1.195 

.218 

1.240 

1.263 

1.288 

«?) 

(C) 

(C) 

(C) 

(C) 

0.030 

0.036 

0.043 

0.051 

0.029 

TABLE  IV—  Continued. 


6 

a=  10° 

a=  11° 

a  =  12° 

a=  13° 

a=  14° 

(#) 

(0) 

CO) 

(£>) 

(-D) 

0 

0.970 

0.964 

0.957 

0.950 

0.942 

5 

1.000 

0.997 

0.993 

0.988 

0.983 

10 

1.031 

1.031 

1.030 

1.028 

1.026 

15 

1.064 

1.067 

1.069 

1.061 

1.072 

20 

1.099 

1.105 

1.110 

1.116 

1.121 

25 

1  .  1  36 

1.147 

1.156 

1.165 

1.173 

30 

1.178 

1.194 

1.204 

1.220 

1.232 

35 

1.224 

1.244 

1.262 

1.281 

1.300 

40 

1.291 

1.304 

1.328 

1.353 

1.377 

45 

1.342 

1.375 

1.407 

1.438 

1.469 

TABLE  V— Continued. 


e 

a=  10° 

a  —  11° 

a=  12° 

a=  13° 

a=  14° 

C0) 

(Jg?) 

CE) 

CE) 

CE) 

0  ^ 

0 

0 

0 

0 

0 

5 

0.030 

0.032 

0.036 

0.039 

0.042 

10 

0.061 

0.067 

0.073 

0.079 

0.085 

15 

0.093 

0.102 

0.111 

0.119 

0.130 

20 

0.124 

0.137 

0.150 

0.163 

0.175 

25 

0.156 

0.173 

0.189 

0.205 

0.221 

30 

0.188 

0.208 

0.216 

0.248 

0.269 

35 

0.220 

0.244 

0  268 

0.292 

0.316 

40 

0.252 

0.280 

0.308 

0.336 

0.365 

45 

0.284 

0.316 

0.349 

0.382 

0.415 

110 


TABLES. 


TABLE  HI— Continued. 


a 

a  =  15° 

a=  16° 

a  =  17° 

a=  18° 

a=  20° 

(#) 

(fi)                  .    f#) 

(B) 

go 

n 

1.035            1.040 

1.048 

1.051 

1.062 

5 

1.059 

1.066 

1.076 

1.081 

1.098 

10 

1.084 

1.093 

1.104 

1.112 

1.132 

15 

1.110 

1.120 

1.134 

1.138 

1.168 

20 

1.135 

1.149 

1.165 

1.177 

1.218 

25 

1.165 

1.179 

1.197 

1.211 

1.245 

30 

1.195 

1.212 

1.233 

1  .  248 

1.288 

35 

1.229 

1.249 

1.272 

1.291 

1.339 

40 

1.268 

1.291 

1.317 

1.340 

1.389 

45 

1.313 

1.338 

1.369 

1.393 

1.451 

(CO 

CO) 

(C) 

(C) 

(CO 

0.067 

0.076 

0  086 

0.095 

0.117 

TABLE  IN— Continued. 


1-0 

a        18° 

6 

a          t 

(i» 

(D) 

cz>) 

(D) 

(0) 

f\ 

0.933 

0.924 

0.915 

0.905 

0.883 

5 

0.977 

0.971 

0.964 

0  957 

0.940 

10 

1.023 

1.018 

1.016 

1.011 

1.000 

15 

1.072 

1.073 

1.071 

1  069 

1.068 

20 

1.124 

1.127 

1.129 

1.131 

1.132 

25 

1.181 

1.188 

1.194 

1.200 

1.208 

30 

1.244 

1.256 

1.266 

1.276 

1.293 

35 

1.316 

1.332 

1.348 

1.363 

1.390 

40 

1.400 

1.422 

1.444 

1.465 

1.505 

45 

1.500 

1.530 

1.559 

1.588 

1.643 

TABLE  N— Continued. 


e 

a=  15° 

a  =  16° 

a=  17° 

a=  18° 

a=  20° 

CE) 

(#) 

CE) 

(#) 

(#) 

0                0 

0 

0 

0 

0 

5 

0.045 

0.047 

0.050 

0.053 

0.058 

10 

0.091 

0.097 

0.102 

0.108 

0.119 

15 

0.139 

0.148 

0.157 

0.165 

0.183 

20 

0.188 

0.200 

0.213 

0.225 

0  249 

25 

0.238 

0  254 

0.270 

0.177 

0  .  31  S 

30 

0  .  289 

0.309 

0  .  3~>9 

0.349 

0.389 

35 

0  341 

0.365 

0.390 

0.414 

0.463 

40 

0.394 

0.42:5 

0.452 

0.481 

0.539 

45 

0.448 

0.482 

0.516 

0.551 

0.620 

TABLE  "VI. 

NATURAL  SINES,  COSINES,  TANGENTS 
AND    COTANGENTS. 


112 


NATDKAL  SINES   AND    COSINES. 


/ 

0° 

1° 

2° 

3« 

40 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

$ 

~o 

.00000 

One. 

.01745 

.99985 

.03490 

.99939 

.05234 

T99863 

.06976 

.99756 

60 

1 

.00029 

One. 

.01774 

.99984 

.03519 

.99938 

.05263 

.99861 

.07005 

.99754 

59 

2 

.00058 

One. 

.01803 

.99984 

.03548 

.99937 

.05292 

.99860 

.07034 

.99752 

58 

3 

.00087 

One. 

.01832 

.99983 

.03577 

.99936 

.05321 

.99858 

.07063 

.99750 

57 

4 

.00116 

One. 

i  .01862 

.99983 

.03606 

.99935 

.05350 

.99857 

.07092 

.99748 

56 

5 

.00145 

One. 

.01891 

.99982 

.03635 

.99934 

.05379 

.99855 

.07121 

.99746 

55 

6 

.00175 

One. 

.01920 

.99982 

.03C64 

.99933 

.05408 

.99854 

.07150 

.99744 

54 

7 

.00204 

One. 

.01949 

.99981 

.03693 

.99932 

.05437 

.99852 

.07179 

.99742 

53 

8 

.00233 

One. 

.01978 

.99980 

.03723 

.99931 

.05466 

.99851 

.07208 

.99740 

52 

9 

.00262 

One. 

.02007 

.99980 

.03752 

.99900 

.05495 

.99849 

.07237 

.99738 

51 

10 

.00291 

One. 

.02036 

.99979 

.03781 

.99929 

.05524 

.99847 

.07266 

.99786 

50 

11 

.00320 

.99999 

.02065 

.99979 

.03810 

.99927 

.05553 

.99846 

.07295 

.99734 

49 

12 

.00349 

.99999 

.02094 

.90978 

.03839 

.99926 

.05582 

.99844 

.07324 

.99731 

48 

13 

.00378 

.99999 

.02123 

.99077 

.03808 

.99925 

.05611 

.99842 

.07353 

.99729 

47 

14 

.00407 

.99999 

.02152 

.99977 

.03897 

.99924 

.05640 

.99841 

.07382 

.99727 

46 

15 

.00436 

.99999 

.02181 

.99976 

.03926 

.99923 

.05669 

.90839 

.07411 

.99725 

45 

16 

.00465 

.99999 

.02211 

.90976 

.03955 

.99922 

.05698 

.90838 

.07440 

.99723 

44 

17 

.00495 

.99999 

.02240 

.99975 

.03984 

.99921 

.05727 

.90836 

.07469 

.99721 

43 

18 

.00524 

.99999 

.02209 

.99974 

.04013 

.99919 

.05756 

.90834 

.07498 

.9P719 

42 

19 

.00553 

.99998 

.02298 

.90974 

.04042 

.99918 

.05785 

.90833 

.07527 

.99716 

41 

20 

.00582 

.99998 

.02327 

.99973 

.04071 

.99917 

.05814 

.99831 

.07556 

.99714 

40 

21 

.00611 

.99998 

.02356 

.99972 

.04100 

.99916 

.05844 

.99829 

.07585 

.99712 

39 

22 

.00640 

.99998 

.02385 

.99972 

.04129 

.99915 

.C5873 

.99827 

.07614 

.99710 

38 

23 

.00669 

.99998 

.02414 

.99971 

.04159 

.99913 

.C5902 

.99826 

.07643 

.99708 

37 

24 

.00698 

.99998 

.02443 

.99970 

.04188 

.99912 

.05931 

.99824 

.07672 

.99705 

86 

25 

.00727 

.99997 

.02472 

.99969 

.04217 

.99911 

.05960 

.99822 

.07701 

.99703 

35 

26 

.00756 

.99997 

.02501 

.99969 

.04246 

.99910 

.05989 

.99821 

.07730 

.99701 

34 

27 

.00785 

.99997 

.02530 

.99968 

.04275 

.99909 

.06018 

.90819 

.07759 

.99699 

33 

28 

.00814 

.99997 

.02560 

.99967 

.04304 

.99907 

.06047 

.99817 

.07788 

.99696 

32 

29 

.00844 

.99996 

.02589 

.99966 

.04333 

.92906 

.06076 

.99815 

.07817 

.99G94 

31 

30 

.00873 

.99996 

.02618 

.99966 

.04362 

.99905 

.06105 

.99813 

.07846 

.99692 

30 

31 

.00902 

.99996 

.02647 

.99965 

.04391 

.99904 

.06134 

.99812 

.07875 

.99689 

29 

32 

.00931 

.99996 

.02676 

.99964 

.04420 

.99002 

.06163 

.99810 

.07904 

.99G87 

28 

33 

.00960 

.99275 

.02705 

.99963 

.04449 

.99901 

.06192 

.99808 

.07933 

.99085 

27 

34 

.00989 

.99995 

.02734 

.99963 

.04478 

.99900 

.06221 

.99806 

.07962 

.99G83 

26 

35 

.01018 

.99995 

.02763 

.99962 

.04507 

.99898 

.06250 

.99804 

.07991 

.99080 

25 

36 

.01047 

.99995 

.02792 

.99961 

.04536 

.99897 

.06279 

.99803 

.08020 

.99078 

24 

37 

.01076 

.99994 

.02821 

.99960 

.04565 

.99896 

.06308 

.99801 

.08049 

.99676 

23 

38 

.01105 

.99994 

.02850 

.99959 

.04594 

.99894 

.06337 

.99709 

.08078 

.99673 

22 

39 

.01134 

.99994 

.02879 

.99959 

.04G23 

.90893 

.06360 

.99797 

.08107 

.99071 

21 

40 

.01164 

.99993 

.02908 

.99958 

.04653 

.99892 

.06395 

.99795 

.08136 

.99668 

20 

41 

.01193 

.99993 

.02938 

.99957 

.04682 

.99890 

.06424 

.99793 

.08165 

.99666 

19 

42 

.01222 

.99993 

.02967 

.99956 

.04711 

.99889 

.06453 

.99702 

.08194 

.99664 

18 

43 

.01251 

.99992 

.02996 

.99955 

.04740 

.99888 

.06482 

.99790 

.08223 

.99661 

17 

44 

.01280 

.99992 

.03025 

.99954 

.04769 

.99886 

.06511 

.99788 

.08252 

.99659 

16 

45 

.01309 

.99991 

.03054 

.90953 

.04798 

.99885 

.06540 

.99786 

.08281 

.99657 

15 

46 

.013381.99991 

.03083 

.99952 

.04827 

.99883 

.06569 

.99784 

.08310 

.99654 

14 

47 

.01367 

.99991 

.03112 

.99952 

.04856 

.99882 

.06598 

.99782 

.08339 

.99652 

13 

48 

.01396 

.99990 

.03141 

.99951 

.04885 

.99881 

.06627 

.99780 

.083G8 

.99649 

12 

49 

.01425 

.99990 

.03170 

.99950 

.04914 

.99879 

.06656 

.99778 

.08397 

.99647 

11 

50 

.01454  .99989 

.03199 

.99949 

.04943 

.99878 

.06685 

.99776 

.08426 

.99644 

10 

51 

.01483  .99989 

.03228 

.99948 

.04979 

.99876 

.06714 

.99774 

.08455 

.99642 

9 

52 

.01513  .99989 

.03257 

.99947 

=05001 

.99875 

.06743 

.99772 

.08484 

.99639 

8 

53 

.01542  .99988 

.03286 

.99946 

.05030 

.99873 

.06773 

.99770 

.08513 

.99637 

7 

54 

.015711.99988 

.03316 

.99945 

.05059 

.99872 

.06802 

.99768 

.08542 

.99035 

6 

55 

.016001.99987 

.03345 

.99944 

.05088 

.99870 

.06831 

.99766 

.08571 

.99632 

5 

56 

.016291.99987 

.03374 

.99943 

.05117 

.99869 

.06860 

.99764 

.08600 

99030 

4 

57 

.01658  .99986 

.03403 

.99942 

.05146 

.99867 

.06889 

.99762 

.08629 

.99627 

3 

58 

.01687  .99986 

.03432 

.99941 

.05175 

.99866 

.06918 

.99760 

.08658 

99025 

2 

59 

.01716  .99985 

.03461 

.99940 

.05205 

.99864 

.06947 

99758 

.08687 

99022 

1 

GO 

.01745 

.99985 

.03490 

.99939 

.05234 

.99863 

.06976 

99756 

.08716 

99619 

_0 

/ 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

t 

-  89» 

»*  88» 

87» 

.  •  86*  -  '• 

85» 

NATURAL  SINES  AND   COSINES, 


113 


6° 

1    6° 

70 

8- 

9° 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

U 

.08716 

.99619 

.10453 

.99452 

.12187 

.99255 

.13917 

.99027 

.15643 

.98769 

60 

l 

.08745 

.99617 

.10482 

.99449 

.12216 

.99251 

.13946 

.99023 

.15672 

.98764 

59 

2 

.08774 

.99614 

.10511 

.99446 

.12245 

.99248 

.13975 

.99019 

.15701 

.98760 

5L!  : 

3 

.08803 

.99612 

.10540 

.99443 

.12274 

.99244 

.14004 

.99015 

.15730 

.98755 

5; 

4 

.08831 

.99609 

.10569 

.99440 

.12302 

.99240 

.14033 

.99011 

.15758 

.98751 

5C 

5 

.08860 

.99607 

.10597 

.99437 

.12331 

.99237 

.14061 

.99006 

.15787 

.98746 

55 

6 

.08889 

.99604 

.10626 

.99434 

.12360 

.99233 

.14090 

.99002 

.15816 

.98741 

54 

7 

.08918 

.99602 

.10655 

.99431 

.12389 

.99230 

.14119 

.98998 

.15845 

.98737 

53 

8 

.08947 

.99599 

.10684 

.99428 

.12418 

.99226 

.14148 

.98994 

.15873 

.98732 

52 

9 

.08976 

.99596 

.10713 

.99424 

.12447 

.99222 

.14177 

.98990 

.15902 

.98728 

51 

10 

.09005 

.99594 

.10742 

.99421 

.12476 

.99219 

.14205 

.98986 

.15931 

.98723 

50 

11 

.09034 

.99591 

.10771 

.99418 

.12504 

.99215 

.14234 

.98982 

.15959 

.98718 

49 

12 

.09003  .90588 

.10800 

.99415 

.12533 

.99211 

.14263 

.98978 

.15988 

.98714 

48 

13 

.09092 

.99586 

.10829 

.99412 

.12562 

.99208 

.14292 

.98973 

.16017 

.98709 

47 

14 

.09121 

.99583 

.10858 

.99409 

.12591 

.99204 

.14320 

.98969 

.16046 

.98704 

46 

15 

.09150 

.90580 

.10887 

.99406 

.12620 

.99200 

.14349 

.989G5 

.16074 

.98700 

45 

16 

.09179 

.99578 

.10916 

.99402 

.12649 

.99197 

.14378 

,98961 

.16103 

.98695 

44 

ir 

.09208 

.99575 

.10945 

.99399 

.12678 

.99193 

.14407 

.98957 

.16132 

.98690 

43 

18 

.09237 

.99572 

.10973 

.99396 

.12706 

.99189 

.14436 

.98953 

.16160 

.98686 

42 

19 

.09266 

.99570 

.11002 

.99393 

.12735 

.99186 

.14464 

.98948 

.16189 

.98681 

41 

20 

.09295 

.99567 

.11031 

.99390 

.12764 

.99182 

.14493 

.98944 

.16218 

.98676 

40 

21 

.09324 

.99564 

.11060 

.99386 

.12793 

.99178 

.14522 

.98940 

.16246 

.98671 

39 

22 

.09353 

.995G2 

.11089 

.99383 

.12822 

.99175 

.14551 

.98936 

.16275 

.98667 

33 

23 

.09382 

.99559 

.11118 

.99380 

.12851 

.99171 

.14580 

.98931 

.16304 

.98662 

37 

24 

.09411 

.99556 

.11147 

.99377 

.12380 

.99167 

.14GC8 

.98927 

.16333 

.98657 

86 

25 

.09440 

.99553 

.11176 

.99374 

.12908 

.99163 

.14637 

.98923 

.16361 

.98652 

35 

26 

.09469 

.99551 

.11205 

.99370 

.12937 

.99160 

.14666 

.98919 

.16390 

.98648 

84 

27 

.09498 

.995-18 

.11234 

.99367 

.12966 

.99156 

.14695 

.98914 

.16419 

.98643 

83 

28 

.09527 

.99545 

.11263 

.99364 

.12995 

.99152 

.14723 

.98910 

.16447 

.98638 

82 

29 

.09556 

.99542 

.11291 

.99380 

.13024 

.99143 

.14752 

.98906 

.16476 

.98633 

81 

30 

.09585 

.99540 

.11320 

.99357 

.13053 

.99144 

.14781 

.98902 

.16505 

.98629 

80 

31 

.09614 

.99537 

.11349 

.99354 

.13081 

.99141 

.14810 

.98897 

.16533 

.98624 

29 

32 

.09642 

.99534 

.11378 

.99351 

.13110 

.99137 

.146C8 

.98893 

.16562 

.98619 

23 

33 

.09671 

.99531 

.11407 

.99347 

.13139 

.99133 

.14867 

.98889 

.16591 

.98614 

27 

34 

.09700 

.99528 

.11436 

.99344 

.13163 

.99129 

.14896 

.98884 

.16620 

.98609 

26 

35 

.09729 

.99526 

.11465 

.99341 

.13197 

.99125 

.14925 

.98880 

.16648 

.98604 

25 

36 

.09758 

.99523 

.11494 

.99337 

.13226 

.99122 

.14954 

.98876 

.16677 

.98600 

24 

37 

.09787 

.99520 

.11523 

.99334 

.13254 

.99118 

.14982 

.98871 

.16706 

.98595 

23 

38 

.09816 

.99517 

.11552 

.99331 

.13283 

.99114 

.15011 

.98867 

.16734 

.98590 

22 

39 

.09845 

.99514 

.11580 

.99327 

.13312 

.99110 

.15040 

.98863 

.16763 

.98585 

21 

40 

.09874 

.99511 

1  .11609 

.99324 

.13341 

.99106 

.15069 

.98858 

.16792 

.98580 

20 

41 

.09903 

.99508 

1  .11638 

.99320 

.13370 

.99102 

.15097 

.98854 

.16820 

.98575 

19 

42 

.09932 

.99506 

.11667 

.99317 

.13390 

.99098 

.15126 

.98849 

.16849 

.98570 

18 

43 

.00961 

.90503 

|  .11696 

.99314 

.13427 

.99094 

.15155 

.98845 

.16878 

.98565 

17 

44 

.00990 

.90500 

.11725 

.99310 

.13450 

.99091 

.15184 

.98841 

.16906 

.98561 

16 

45 

.10019 

.99497 

.11754 

•99307 

.13485 

.99087 

.15212 

.98836 

.16935 

.98556 

15 

46 

.10048 

.99494 

.11783 

.99303 

.13514 

.99083 

.15241 

.98832 

.16964 

.98551 

14 

47 

.10077 

.99491 

.11812 

.99300 

.13543 

.99079 

.15270 

.98827 

.16992 

.98546 

13 

48 

.10106 

.99488 

.11840 

.99297 

.13572 

.99075 

.15299 

.98823 

.17021 

.98541 

12 

49 

.10135 

.99485 

.11869 

.99293 

.13600 

.99071 

.15327 

.98818 

.17050 

.98536 

11 

50 

.10164 

.89482 

.11898 

.99290 

.13629 

.99067 

.15356 

.98814 

.17078 

.98531 

10 

51 

.10192 

.99479 

.11927 

.99286 

.13658 

.99063 

.15385 

.98809 

.17107 

.98526 

9 

52 

.10221 

.99476 

.11956 

.99283 

.13637 

.99059 

.15414 

.98805 

.17136 

.98521 

8 

53 

.10250 

.99473 

.1x985 

.99279 

.13716 

.99055 

.15442 

.98800 

.17164 

.98516 

7 

54 

.10279 

.99470 

.12014 

.99276 

.13744 

.99051 

.15471 

.98796 

.17193 

.98511 

6 

55 

.10308 

.99467 

.12043 

.99272 

.13773 

.99047 

.15500 

.98791 

.17222 

.98506 

5 

56 

.10337 

.99464 

.12071 

.99269 

.13802 

.99043 

.15529 

.98787 

.17250 

.98501 

4 

57 

10366 

.99461 

.12100 

.99265 

.13831 

.990°,9 

.15557 

.98782 

.17279 

.98496 

3 

58 

.10395 

.99458 

.12129 

.99262 

.13860 

.99035 

.15586 

.98778 

.17308 

.98491 

2 

59 

.10424 

.99455 

.12158 

.99258 

.13889 

.99031 

.1*615 

.98773 

.17336 

.98486 

1 

60 

.10453 

.99452 

.12187 

.99255 

.13917 

.99027 

.15643  .98769 

.17365 

.98481 

J) 

0 

Cosin 

Sine 

Cosin 

Sine" 

Cosin 

Sine 

Cosin  j  Sine 

Cosin 

Sine 

t 

84* 

83' 

83° 

81° 

80° 

114 


NATURAL  SINES  AND  COSINES. 


10° 

11° 

12° 

13° 

14» 

/ 

Sine  Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

i 

"o 

.17365"!.  98481 

.19081 

.98163 

.20791 

.97815 

.22495 

797437 

.24192 

.97030 

60 

I 

.17393  '.98476 

.19109 

.98157 

.20820 

.97809 

.22523 

.97430 

.24220 

.97023 

59 

2 

.17422  .98471 

.19138 

.98152 

.20848 

.97803 

.22552 

.97424 

.24249 

.97015 

58 

3 

.17451  ..98466 

.19167 

.98146 

.20877 

.97797 

.22580 

.97417 

.24277 

.97008 

57 

1 

.17479  .93461 

.19195 

.98140 

.20905 

.97791 

.22608 

.97411 

.24305 

.97001 

56 

5 

.17508 

.98455 

.19224 

.98135 

.20933 

.97784 

.22637 

.97404 

.24333 

.96994 

55 

6 

.17537 

.98450 

.19252 

.98129 

.23962 

.97778 

.22665 

X)7398 

.24362 

.96987 

54 

7 

17565 

.98445 

.19281 

.98124 

.20990 

.97772 

.22693 

.97391 

.24390 

.96980 

53 

8 

.17594 

.93440 

.19309 

.98118 

.21019 

.97766 

.22722 

.97384 

.24418 

.96973 

52 

9 

.17623 

.93435 

.19338 

.98112 

.21047 

.97760 

.22750 

.97378 

.24446 

.96966 

51 

10 

.17651 

.98430 

.19366 

.98107 

.21076 

.97754 

.22778 

.97371 

.24474 

.96959 

50 

11 

.17680 

.98425 

.19395 

.98101 

.21104 

.97748 

.22807 

.97365 

.24503 

.96952 

49 

12 

.17708 

.CJ1CO 

.19423 

.93336 

.21132 

.97742 

.223.35 

.97358 

.24531 

.96945 

48 

13 

.17737 

.98414 

.19452 

.93C90 

.21101 

.97735! 

.22363 

.97351 

.24559 

.96937 

47 

14 

.17768 

.93409 

.19481 

.93384 

.21189 

.97729 

.22892 

.97345 

.24587 

.96930 

46 

15 

.17794 

.93404 

.19509 

.98379 

.21218 

.97723! 

.22920 

.97338 

.24615 

.96923 

45 

15 

.17823 

.93339 

.19538 

.93373 

.2124G 

.97717 

.22948 

.97331 

.24644 

.96916 

44 

1? 

.17852 

.98394 

.19503 

.93087 

.21275 

.97711 

.22977 

.97325 

.24672 

.96909 

43 

IS 

.17880 

.98389 

.19595 

.93361 

.21303 

.97705 

.23005 

.97318 

.24700 

.96902 

42 

13 

.17909 

.98333 

.19623 

.98356 

.21331 

.97698 

.23033 

.97311 

.24728 

.96894 

41 

20 

.17937 

.98378 

.19652 

.98050 

.21360 

.97692 

.23062 

.97304 

.24756 

.96887 

40 

21 

.17966 

.98373 

.19680 

.98044 

.21388 

.97686 

.23090 

.97298 

.24784 

.96880 

39 

22 

17995 

.93368 

.19703 

.93339 

.21417 

.97680! 

.23118 

.97291 

.24813 

.96873 

38 

2i 

.18023 

.93302 

.19737 

.93333 

.214  13 

.97673 

.23146 

.97284 

.24841 

.96866 

37 

£1 

18052 

.98357 

.19766 

93327 

.21474 

.97667 

.23175 

.97278 

.24869 

.96858 

36 

23 

18081 

.98352 

.19794 

.93321 

.21502 

.97661 

.23203 

.97271 

.24897 

.96851 

35 

23 

18109 

.93347 

.19323 

93316 

.21533 

.97655 

.23231 

.97264 

1.24925 

.96844 

34 

27 

10138 

.98341 

.19851 

98310 

.21559 

.97648 

.23200 

.97257 

.24954 

.96837 

33 

23 

18166 

.98336 

.18880 

93004 

.21537 

.97642 

.23283 

.97251 

.24982 

.96829 

32 

23 

10195 

.93331 

.19903 

97938 

.21C1G 

.97030! 

.23316 

.97244 

.25010 

.96822 

31 

30 

18224 

.98325 

.19937 

97992 

.21644 

.97630 

.23345 

.97237 

.25038 

.96815 

30 

31 

18252 

.98320 

.19965 

97987 

.21072 

97623  ' 

.23373 

.97230 

.25066 

.96807 

29 

18281 

.93315 

.19394 

97331 

.21701 

.97017 

.23401 

.97223 

.25094 

.96800 

28 

33 

18309 

.93310 

.20022 

97975 

.21723 

97G11 

.23423 

.97217 

.25122 

.96793 

27 

34 

18338 

.93304 

.20051 

97969 

.21753 

97G34 

.23453 

.97210 

.25151 

.96786 

26 

35 

18367 

.98299 

.20379 

97963 

.21733 

3^533 

.23483 

.97203 

.25179 

.96778 

25 

30 

18395 

.93294 

.23108 

97958 

.21014 

97592 

.23514 

.97196 

.25207 

.96771 

24 

37 

18424 

.98288 

.20136 

97952 

.21043 

97585 

.23542 

.97189 

.25235 

.96764 

23 

38 

18452 

.98283 

.20165 

97946 

.21871 

97579 

.23571 

.97182 

.25263  .96756 

22 

39 

18481 

.93277 

.20193 

97940 

.21899 

97573 

.23593 

.97176 

.25291 

.96749 

21 

40 

18509 

.98272 

.20222 

97934 

.21928 

97566 

.23627 

.97169 

.25320 

.96742 

20 

41 

18538 

.98267 

.20250 

97928 

.21956 

97560 

.23656 

.97162 

.25348 

.96734 

19 

42 

18567 

.93201 

.20279 

97922 

.21985 

97553 

.23684 

.97155 

.25376 

.96727 

18 

43 

18595 

.93256 

23307 

97916 

.22013 

97547 

.23712 

.97148 

.25404 

.96719 

17 

44 

18624 

.93250 

.'20336 

97910 

.22041 

.97541 

.23740 

.97141 

.25432 

.96712 

16 

45 

18052 

.93245 

.20364 

97905 

.22070 

.97534 

.23769 

.97134 

.25460 

.96705 

15 

4G 

1GG81 

.93240 

.20393 

97899 

.22098 

.97528 

.23797 

.97127 

.25488 

.96697 

14 

47 

1G710 

.93234 

.20421 

97893 

.22126 

.97521 

.23825 

.97120 

.25516 

.96690 

13 

43 

.10738 

.93229 

.20450 

97887 

.22155 

.97515 

.23853 

.97113 

.25545 

.96682 

12 

49 

.1G7G7 

.93223 

.20478 

97881 

.22183 

.97508 

.23882 

.97106 

.25573 

.96675 

11 

50 

.10795 

.93218 

.20507 

97875 

.22212,.  97502 

.23910 

.97100 

.25601 

.96667 

10 

51 

.10824 

.93212 

.23535 

97869 

.22240  '.97490 

.23938 

.97093 

.25629 

.96660 

9 

52 

.18852 

.93207 

.20563 

97863 

.  22268  |.  97489 

.23966 

.97086 

.25657  .96653 

8 

53 

.18881 

.93201 

.20592 

.97857 

.22297 

.97483 

.23995 

.97079 

.25685  .96645 

7 

54 

.10910 

.98190 

.20620 

.97851 

.22325 

.97476 

.24023 

.97072 

.25713 

.96638 

6 

55 

.18938 

.98190 

.2C349 

.97845 

.22353 

.97470 

.24051 

.97065 

.25741 

.96630 

5 

56 

18967 

.98185 

.20677 

.97839 

.22382 

.97463 

.24079 

.97058 

.25769  .96623 

4 

57 

18995 

.98179 

.20706 

.97833 

.22410 

:  97457 

.24108 

.97051 

.25798  .96615 

3 

58 

.19024 

.98174 

-.20734 

.97827 

.22438 

.97450 

.24136 

.97044 

.25826  .96608 

2 

59 

.19052 

.98168 

.20763 

.97821 

.22467 

.97444 

.24164 

.97037 

.25854  .96600 

1 

GO 

.19081 

.98163 

.20791 

.97815 

.22495 

.97437 

.24192 

.97030 

.258821.96593 

•  l,  — 

0 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin  I  Sine 

/ 

79° 

78°    ' 

77°  ' 

76° 

75" 

NATURAL   SINES    AND    COSINE'S. 


115 


21 


31 


15° 


Sine 


.25882 
.25910 
.25938 
.25966 
.25994 
.26022 
.26050 
.26079 
.26107 
.26135 


,26191 


.26247 
.26275 


Cosin 


.96593 
.96585 
.98678 
.96570 
.96562 
.96555 
.96547 
.96540 


.96517 


.26331 
,26359 


.96494 
.96486 
.96479 
.96471 
.96463 
.96456 
.96448 
.96440 

.26471  .96433 
26500  .96425 
26528  .96417 
26556  .96410 


,26415 


.96612 
.26640 


.26696 
.26724 

.26752 

.26780 


.26948 
.26976 
.27004 

.27032 
.27060 
.27088 
.27116 
.27144 
.27172 
.27200 
.27228 
.27256 


.96386 
.96379 
.96371 


.96355 
.96347 
.96340 
.96332 
.96324 
.96316 
.96308 
.96301 
.96293 


.96277 


.96261 
.96253 
.96246 


.27312 
.27340 
.27368 
.27396, 

.27424 
.27452 
.27480! 
.275081 
.275361 
.27564 


.96230 
.96222 
.96214 
.96206 

.96198 
.96190 
.96182 
.96174 


.96158 
.96150 
.96142 
.96134 


Cosin 


Sine 


74° 


16' 


Sine  Cosin 


.27564  .96126 
.27592  .96118 
.27620  .96110 
.27648  .96102 
.276761.96094 
.27704 !. 96086 
.277311.96078 


17" 


Sine 


.29237 


.27759 
.27787 
.27815 
.27843 

.27871 


.27927 
.27955 

.27983 
.28011 


.28067 
.28095 
.28123 

.28150 

.28178 
.28206 


.28262 
.28290 
.28318 


.28374 


.2845^ 


.28513 
.28541 
.28569 
.28597 


.28708 
.28736 
.28764 
.28792 


.28875 


•96070 


.96054 


.96037 


.96013 
.96005 
.95997 


.95981 
.95972 
.95964 

.95956 


.95940 
.95931 
.95923 
.95915 
.95907 
.95898 
.95890 


.95874 

.95865 
.95857 


.95841 
.95832 
.95824 
.95816 
.95807 


.95791 
.95782 
.95774 
.95766 
.95757 
.95749 
.95740 
.95732 
.95724 
.95715 

.95707 

.95698 


.95681 
,95673 
,95664 
,95656 
,95647 


.29237 


Cosin 


.95630 
Sine 


.29321 
.29348 
.29376 
.29404 
.29432 
.29460 
.29487 
.29515 

.29543 

.29571 
.29599 
.29626 
.29654 
tB9682 
.29710 
.2973' 
.29765 
.29793 

.29821 

.29849 
.29876 


.29960 
.29987 
.30015 
.30043 
.30071 


.30126 
.30154 


.30209 

.30237 


.30320 
.30348 

.30376 


.30431 
.30459 
.30486 
.30514 
.30542 
.30570 
.30597 


.30708 


.30763 
.30791 
.30819 
.30846 
.30874 
.30902 


Osin 


Cosin 


.95630 


95613 
95605 


18° 


Sine 


95588 
95579 
95571 
95562 
95554 
,95545 

95536 

95528 
95519 
95511 
95502 
95493 
95485 
95476 
95467 
95459 

95450 
,95441 
95433 
,95424 
,95415 
,95407 


.95380 
,95372 


.95354 
.95345 
.95337 


.95319 
.95310 

.95301 
.95203 


.95275 

.95266 
.95257 
.95248 


.95231 
.95222 
.95213 
.95204 
.95195 

.95186 
.95177 
.95168 
.95159 
.95150 
.95142 
.95133 
.95124 
.95115 
.95106 
Sine 


.30957 
.30985 
.31012 
.31040 
.31068 
.31095 
.31123 
.31151 
.31178 

.31206 

.31233 
.31261 
.31289 
.31316 
.31344 
.31372 
.31399 
.31427 
.31454 

.31482 
.31510 
.31537 
.31565 
.31593 
.31620 
.31648 
.31675 
.31703 
.31730 

.31758 
.31786 
.31813 
.31841 
.31868 
.31896 
.31923 
.31951 
.31979 
.82006 

.32034 

.32061 
.32089 
.32116 
.32144 
.32171 
.32199 
.32227 
.32254 


.32419 

.32447 
.32474 


72' 


.32529 
.32557 
Cosin 


Cosin 


.95106 
.95097 
.95088 
.95079 
.95070 
.95061 
.95052 
.95043 
.95033 


.95015 

,95006 
,94997 


.94979 
.94970 
.94961 
.94952 
.94943 
.94933 


.94915 

.94906 
.94897 
.94888 
.94878 
.94869 
.94860 
.94851 
.94842 
.94832 


.94814 
.94805 
.94795 
.94786 
.94777 
.94768 
.94758 
.94749 
.94740 

.94730 
.94721 
.94712 
.94702 


.94684 

,94674 
,94665 
,94656 
,94646 

94637 

94627 
94618 


.94590 


94571 
94561 
94552 


Bind 


71° 


19° 


Sine 


.32557 
.32584 
.32612 


.32667 
.32694 
.32722 
.32749 
.32777 
.82804 


.32859 


.32914 
.32942 
.32969 
.32997 
.33024 
.33051 
.83079 
.83106 

.83134 

.33161 
.83189 
.33216 
.83244 
.83271 
.83298 


.83353 


.33408 

.83436 

.33463 

.33490 

.33518 

.33545 

.835 

.83600 

.33627 

.83655 


.33710 
.33737 
.33764 
.83792 
.83819 
.83846 
.83874 


8392Q 


.84011 


.84065 


.84120 
.84147 
.84175 
.34202 
Cosin 


Cosin 
194552 
.94542 
.94533 
.94523 
.94514 
.94504 
.94495 
.94485 
.94476 
.94466 
.94457 

.94447 
.94438 
.94428 
.94418 
.94409 
.94399 
.94390 
.94380 
.9437( 
.94361 

.94351 
.94342 
.94332 
.94322 
.94313 
.94303 
.94293 
.94284 
.94274 
.94264 

.94254 
.94245 
.94235 
.94225 
.94215 
.9420( 
.9419( 
.9418( 
.94176 
.94167 

.94157 
.94147 
.94137 
.94127 
.94118 
.94108 
.94098 
.94088 
.94078 
.94068 

.94058 

.94049 
.94039 
,94029 
94019 


93979 
93969 
Sine 


70e 


60 


116 


NATURAL  SINES   AND   COSINES. 


20° 


Cosin 


34257 
34284 
34311 
34339 


34393 
34421 
34448 
34475 

34503 

.34530 
34557 
34584 
34612 
34639 


34694 
.34721 

34748 

34775 
.34803 
34830 
.34857 


34S12 


34966 


.35048 
.35075 
.35102 
.35130 
.35157 
.35184 


.93929 
.93919 
.93909 


.93879 


93829 
93819 


93799 


93779 


93759 

93748 
93738 
,93728 
93718 
93708 


936^ 


.9365? 
.93647 
93637 


21° 


Sine 

135837 

,35864 


Cosin 


.93348 


,35918!. 93327 
.93316 
,35973 

,36000 


,36054 


,36135 
,36162 
,36190 
,36217 
,36244 


36298 
36325 
36352 
36379 


,36434 


,36515 
,36542 
,36569 
,36596 


.36677 
.36704 
.36731 

,36758 


,36812 


.93295 


.93274 


93243 


93211 
,93201 
,93190 
,93180 
,93169 
,93159 
,93148 

.93137 
.93127 
.93116 
.93106 

.93095 
.93084 
.93074 
.93063 


.93043 


.93020 
.93010 


.92978 


22° 


24° 


.40806 
.40833 
.40860 


.37757 
.37784 
.37811 
.37838 
.37865 
.37892 
.37919 
.37946 
.37973 
.37999 


.93488 
.92477 
.92466 
.92455 
.92444 
.92432 
.92421 
.92410 
.92399 


<;<> 


.38322 
.38349 
.38376 
.38403 
.38430 


or 

38 

.OOZ.ll 

.35239 

.  yaoyo 
.93585 

.  ooooy 
.36867 

tweaot 

.92956 

.oo<±yu 
.38483 

.JJ/40.1U 

.92299 

.«±wu« 
.40088 

.WlU^iU 

.91613 

.<±1UOU 

.41681 

.»U»11 

.90899 

«o 

22 

39 

.35266 

.93575 

.36894 

.92945 

.38510 

.92287 

.40115 

.91601 

.41707 

.90887 

21 

40 

.35393 

.93565 

.36931 

.92935 

.38537 

.92376 

.40141 

.91590 

.41734 

.90875 

20 

41 

.35320 

.93555 

.36948 

.92924 

.38564 

.92265 

.40168 

.91578 

.41760 

.90863 

19 

42 

.35347 

.93544 

.36975 

.92913 

.38591 

.92254 

.40195 

.91566 

.41787 

.90851 

18 

43 

.35375 

.93534 

.37002 

.92902 

.88617 

.92243 

.40221 

.91555 

.41813 

.90839 

17 

44 

.35402 

.93524 

.S7029 

.92892 

.38644 

.92231 

.40248 

.91543 

.41840 

.90826 

16 

45 

.35429 

.93514 

.37056 

.92881 

.38671 

.92220 

.40275 

.91531 

.41866 

.90814 

15 

46 

.35456 

.93503 

.37083 

.92870 

.38698 

.92209 

.40301 

.91519 

.41892 

.90802- 

14 

47 

.35484 

.93493 

.37110 

.92859 

.38725 

.92198 

.40328 

.91508 

.41919 

.90790 

13 

48 

.35511 

.93483 

.37137 

.92849 

.38752 

.92186 

.40355 

.91496 

.41945 

.90778 

12 

49 

.35538 

.93472 

.37164 

.92838 

.38778 

.92175 

.40381 

.91484 

.41972 

.90766 

11 

50 

.35565 

.93462 

.37191 

.92837 

.38805 

.92164 

.40408 

.91473 

.41998 

.90753 

10 

51 

.35592 

.93452 

.37218 

.92816 

.38832 

.92152 

.40434 

.91461 

.42024 

.90741 

9 

52 

.35619 

.93441 

.37245 

.92805 

.38859 

.92141 

.40461 

.91449 

.42051 

.90729 

8 

53 

.35647 

.93431 

.37272 

.92794 

.38886 

.92130 

.40488 

.91437 

.42077 

.90717 

7 

54 

.35674 

.93420 

.37299 

.92784 

.38912 

.92119 

.40514 

.91425 

.42104 

.90704 

6 

55 

.35701 

.93410 

.37326 

.92773 

.38939 

.92107 

.40541 

.91414  .42130 

.90692 

5 

56 

.35728 

.93400 

.37353 

.92762 

.38966 

.92096 

.40567  '.91402  .42156 

.90680 

4 

57 

.35755 

.93389 

.37380 

.92751 

.38993 

.92085 

.40594  .91390:  .42183 

.90668 

3 

58 

.35782 

.93379 

.37407 

.92740 

.39020 

.92073 

.406211.91378! 

.42209 

.90655 

2 

59 

.35810 

.93368 

.37434 

.92729 

.39046 

.92062 

.40647  .91366 

.42235 

.90643 

1 

CO 

.35837 

.933581 

.37461 

.92718 

.39073 

.92050 

.406741.91355 

.42262 

.90631 

0 

t 

Cosin 

Sine 

Cosin 

Bine 

Cosin 

Sine 

Cosin  Sine 

Cosin 

Sine 

/ 

69"   1 

68° 

67° 

66' 

65° 

Sine  J  Cosin 
740674  791355 
.407001.91343  59 
.407271.91331  58 
.40753  .91319*57 
.40780  .91307  56 


.91648 
.91636 


.91295 
.91283 
.91272 


.408861.91260 
.409131.91248 
.40939  .91236 


.40966 
.40992 
.41019 
.41045 
.41072 
.41098 
.41125 
.41151 
.41178 
.41204 


.41496 

.41522 
.41549 
.41575 
.41602 
.41628 


91224 
.91212 
.91200 
91188 
.91176 
§1164 
91152 
91140 
91128 
91116 

91104 
91092 
,91080 
91068 
91056 
,91044 
.91032 
,91020 
,91008 


.90972 


.90936 


NATURAL   SINES  AND   COSINES. 


117 


25° 

26° 

27° 

28° 

29° 

9 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

To 

.42262 

.90631 

.43837 

.89879 

.45399 

.89101 

.46947 

.88295 

.48481 

.87462 

60 

l 

.42288 

.90618 

.43863 

.89867 

.45425 

.89087 

.46973 

.88281 

.48506 

.87448 

59 

2 

.42315 

.90606 

.43889 

.89854 

.45451 

.89074 

.46999 

.88267 

.48532 

.87434 

53 

3 

.42341 

.90594 

.43916 

.89841 

.45477 

.89061 

.47024 

.88254 

.48557 

.87420 

57 

4 

.42367 

.90582 

.43942 

.89828 

.45503 

.89048 

.47050 

.88240 

.48583 

.87406 

56 

5 

.42394 

.90569 

.43968 

.89816 

.45529 

.89035 

.47076 

.88226 

.48608 

.87391 

55 

6 

.42420 

.90557 

.43994 

.89803 

.45554 

.89021 

.47101 

.88213 

.48634 

.87377 

54 

7 

.42446 

.90545 

.44020 

.89790 

.45580 

.89008 

.47127 

.88199 

.48659 

.87363 

53 

8 

.42473 

.90532 

.44046 

.89777 

.45606 

.88995 

.47153 

.88185 

.48684 

.87349 

52 

9 

.42496 

.90520 

.44072 

.89764 

.45632 

.88981 

.47178 

.88172 

.48710 

.87335 

51 

10 

.42535 

.90507 

.44098 

.89752 

.45658 

.88968 

.47204 

.88158 

.48735 

.87321 

50 

11 

.42552 

.90495 

.44124 

.89739 

.45684 

.88955 

.47229 

.88144 

.48761 

.87306 

49 

12 

.42578 

.90483 

.44151 

.89726 

.45710 

.88942 

.47255 

.88130 

.48786 

.87292 

48 

13 

.42604 

.90470 

.44177 

.89713 

.45736 

-.88928 

.47281 

.88117 

.48811 

.87278 

47 

14 

.42631 

.90458 

.44203 

.89700 

.45762 

.88915 

.47306 

.88103 

.48837 

.87264 

46 

15 

.42657 

.90446 

.44229 

.89687 

.45787 

.88902 

.47332 

.88089 

.48862 

.87250 

45 

16 

.42683 

.90433 

.44255 

.89674 

.45813 

.88888 

.47358 

.88075 

.48888 

.87235 

44 

17 

.42709 

.90421 

.44281 

.89662 

.45839 

.88875 

.47383 

.88062 

.48913 

.87221 

43 

18 

.42736 

.90403 

.44307 

.89649 

.45865 

.88862 

.47409 

.88048 

.48938 

.87207 

42 

19 

.42762 

.90396 

.44333 

.89636 

.45891 

.88848 

.47434 

.88034 

,48964 

.87193 

41 

20 

.42788 

.90383 

.44359 

.89623 

.45917 

.88835 

.47460 

.88020 

.48989 

.87178 

40 

21 

.42815 

.90371 

.44385 

.89610 

.45942 

.88822 

!l7486 

.88006 

.49014 

.87164 

39 

22 

.42841 

.90358 

.44411 

.89597 

.45968 

.88808 

.47511 

.87993 

.49040 

.87150 

38 

23 

.42367 

.90346 

.44437 

.89584 

.45994 

.83795 

.47537 

.87979 

.49065 

.87136 

37 

24 

.42894 

.90334 

.44464 

.89571 

.4G020 

.83782 

.47562 

.87965 

.49090 

.87121 

36 

25 

.42920 

.90321 

.44490 

.89558 

.46046 

.88768 

.47588 

.87951 

.49116 

.87107 

35 

26 

.42946 

.90309 

.44516 

.89545 

.46072 

.83755 

.47614 

.87937 

.49141 

.87093 

34 

27 

.42972 

.90206 

.44542 

.89532 

.46097 

.88741 

.47639 

.87923 

.49166 

.87079 

33 

23 

.42999 

.90284 

.44568 

.89519 

.46123 

.88728 

.47665 

.87909 

.49192 

.87064 

32 

29 

.43025 

.90271 

.44594 

.89506 

.46149 

.88715 

.47690 

.87896 

.49217 

.87050 

31 

30 

.43051 

.90259 

.44620 

.89493 

.46175 

.88701 

.47716 

.87882 

.49242 

.87036 

30 

31 

.43077 

.90246 

.44646 

.89480 

.46201 

.88688 

.47741 

.87868 

.49268 

.87021 

29 

32 

.43104 

.90233 

.44672 

.89467 

.40226 

.88674 

.47767 

.87854 

.49293 

.87007 

28 

33 

.43130 

.90221 

.44698 

.89454 

.46252 

.88661 

.47793 

.87840 

.49318 

.86993 

27 

34 

.43156 

.90208 

.44724 

.89441 

.46278 

.88647 

.47818 

.87826 

.49344 

.86978 

26 

35 

.43182 

.90196 

.44750 

.89428 

.46304 

.88634 

.47844 

.87812 

.49369 

.86964 

25 

36 

.43209 

.90183 

.44776 

.89415 

.46330 

.88620 

.47869 

.87798 

.49394 

.86949 

24 

37 

.43235 

.90171 

.44802 

.89402 

.46355 

.88607 

.47895 

.87784 

.49419 

.86935 

23 

38 

.43261 

.90158 

.44828 

.89389 

.46381 

.88593 

.47920 

.87770 

.49445 

.86921 

22 

39 

.43287 

.90146 

.44854 

.89376 

.46407 

.88580 

.47946 

.87756 

.49470 

.86906 

21 

40 

.43313 

.90133 

.44880 

.89363 

.46433 

.88566 

.47971 

.87743 

.49495 

.86892 

20 

41 

.43340 

.90120 

.44906 

.89350 

.46458 

.88553 

.47997 

.87729 

.49521 

.86878 

19 

42 

.43366 

.90108 

.44932 

.89337 

.46484 

.88539 

.48022 

.87715 

.49546 

.86863 

18 

43 

.43392 

.90095 

.44958 

.89324 

.46510 

.88526 

.48048 

.87701 

.49571 

.86849 

17 

44 

.43418 

.90082 

.44984 

.89311 

.46536 

.88512 

.48073 

.87687 

.86834 

16 

45 

.43445 

.90070 

.45010 

.89298 

.46561 

.88499 

.4809J 

.87673 

49622 

.86820 

15 

46 

.43471 

.90057 

.45036 

.89285 

.46587 

.88485 

.48124 

.87659 

.'49647 

.86805 

14 

47 

.43497 

.90045 

.45062 

.89272 

.46613 

.88472 

.48150 

.87645 

.49672 

.86791 

13 

48 

.43523 

.90032 

.45088 

.89259 

.46639 

.88458 

.48175 

.87631 

.49697 

.86777 

12 

49 

.43549 

.90019 

.45114 

.89245 

.46664 

.88445 

.48201 

.87617 

.49723 

.86762 

11 

50 

.43575 

.90007 

.45140 

.89232 

.46690 

.88431 

.48226 

.87603 

.49748 

.86748 

10 

51 

.43602 

.89994 

.45166 

.89219 

.46716 

.88417 

.48252 

.87589 

.49773 

.86733 

9 

52 

.43628 

.89981 

.45192 

.89206 

.46742 

.88404 

.48277 

.87575 

.49798 

.86719 

8 

53 

.43654 

.89968 

.45218 

.89193 

.46767 

.88390 

.48303 

.87561 

.49824 

.86704 

7 

54 

.43680 

.89956 

.45243 

.89180 

.46793 

.88377 

.48328 

.87546 

.49849 

.86690 

6 

55 

.43706 

.89943 

.45269 

.89167 

.46819 

.88363 

.48354 

.87532 

.49874 

.86675 

5 

56 

.43733 

.89930 

.45295 

.89153 

.46844 

.88349 

.48379 

.87518 

.49899 

.86661 

4 

57 

.43759 

.89918 

.45321 

.89140 

.46870 

.88336 

.48405 

.87504 

.49924 

.86646 

3 

58 

.43785 

.89905 

.45347 

.89127 

.46896 

.88322 

.48430 

.87490 

.49950 

.86632 

2 

59 

.43811 

.89892 

.45373 

.89114 

.46921 

.88308 

.48456 

.87476 

.49975 

.86617 

1 

60 

.43837 

.89879 

.45399 

.89101 

.46947 

.88295 

.48481 

.87462 

.50000 

.86603 

_0 

t 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

/ 

64° 

63° 

62° 

61°  ' 

60°   1 

118 


NATURAL  SINES  AND  COSINES. 


30° 

31* 

32° 

33° 

34' 

/ 

Sine 

3osin 

Sine 

Cosin 

Sine  I  Cosin 

Sine 

Cosin 

Sine 

Cosin 

/ 

~o 

.50000 

.86003 

.51504 

.85717 

.52992  .84805 

.54464 

.83867 

.55919 

.82904 

60 

1 

.50025 

.86588 

.51529 

.85702 

.53017 

.84789 

.54488 

.83851 

.55943 

.82887  59 

2 

.50050 

.86573 

.51554 

.85687 

.53041 

.84774 

.545131.83835 

.55968 

.82871  58 

3 

.50076 

.86559 

.51579 

.85672 

.53066 

.84759 

.54537 

.83819 

.55992 

.82855  57 

4 

.50101 

86544 

.51604 

.85657 

.53091 

.84743 

.54561 

.8C804 

.56016 

.82839  56 

5 

.50126 

86530 

.51628 

.85642 

.53115 

.84728 

.54586 

.83788 

.56040 

.82822  55 

6 

.50151 

.86515 

.51653 

.85627 

.53140 

.84712 

.54610 

.83772 

.56064 

.82806  54 

7 

.50176 

.86501 

.51678 

.85612 

.53164 

.84697 

.54635 

.83756 

.56088 

.82790  53 

8 

.50201 

.86486 

.51703 

.85597 

.53189 

.84681  ' 

.54659 

.83740 

.56112 

.82773  52 

9 

.50227 

.86471 

.51728 

.85582 

.53214 

.84666 

.54683 

.83724 

.56136 

.82757 

51 

10 

.50252 

.86457 

.51753 

.85567 

.53238 

.84650 

.54708 

.83708 

.56160 

.82741 

50 

11 

.50277 

.86442 

.51778 

.85551 

.53263 

.84635 

.54732 

.83692 

.56184 

.82724 

49 

12 

.50302 

.86427 

.51803 

.85536 

.53288 

.84619 

.54756 

.83076 

.56208 

.82708'  43 

13 

.50327 

.86413 

.51828 

.85521 

.53312 

.84604 

.54781 

.83660 

.56232 

.82392  '  47 

14 

.50352 

.86398 

.51852 

.85506 

.53337 

.84588 

.54805 

.83645 

.56256 

.82675  43 

15 

.50377 

.86384 

.51877 

.85491 

.53361 

.84573 

.54829 

.83029 

.56280 

.82659  43 

16 

.50403 

.86369 

.51902 

.85476 

.53386 

.84557 

.54854 

.83013 

.56305 

.82643144 

-17 

.50428 

.86354 

.51927 

.85461 

.53411 

.84542 

.54878 

.83597 

.56329 

.82626  43 

18 

.50453 

.86340 

.51952 

.85446 

.53435 

.84526 

.54902 

.83581 

.56353 

.82610  42 

19 

.50478 

.86325 

.51977 

.85431 

.53460 

.84511 

.64927 

.83505 

.56377 

.82593  41 

20 

.50503 

.86310 

.52002 

.85416 

.53484 

.84495 

.64951 

.83549 

.56401 

.82577 

40 

2f 

.50528 

86295 

.52026 

.85401 

.53509 

.84480 

.54975 

.83533 

.56425 

.82561 

89 

22 

.50553 

.86281 

.52031 

.83335 

.53534 

.84464 

.54999 

.83517 

.56449 

.82544 

38 

23 

.50578 

.86266 

.52076 

.85370 

.53558 

.84448 

.55024 

.83501 

.56473 

.82528 

37 

24 

.50603 

.86251 

.52101 

.85355 

.53583 

.84433 

.55048 

.83485 

.56497 

.82511 

36 

25 

.50628 

.86237 

.52126 

.83340 

.53607 

.84417 

.55072 

.83469 

.56521 

.82495 

35 

26 

.50654 

.86222 

.52151 

.85325 

.53032 

.84402 

.55097 

.83453 

.56545 

.82478 

34 

27 

.50679 

.86207 

.52175 

.85310 

.53056 

.84386 

.55121 

.83437 

.56509 

.82462 

33 

28 

.50704 

.86192 

.52200 

.85294 

.53681 

.84370 

.55145 

.83421 

.56593 

.82446 

32 

29 

.50729 

.86178 

.52225 

.85279 

.53705 

.84355 

.55169 

.83405 

.56617 

.82429 

31 

80 

.50754 

.86163 

.52250 

.85264 

.53730 

.84339 

.55194 

.83389 

.56641 

.83413 

36 

31 

.60779 

.86148 

.52275 

.85249 

.53754 

.84324 

.55218 

.83373 

.56665 

.82396 

29 

32 

.50804 

.86133 

.52293 

.83234 

.53779 

.84308 

.55242 

.83356 

.56689 

.82380 

28 

83 

.50829 

.86119 

.52324 

.85218 

.53804 

.84292 

.55206 

.83340 

.56713 

.82363 

27 

34 

.50854 

.86104 

.52349 

.85203 

.53828 

.84277 

.55291 

.83324 

.56736 

.82347 

26 

35 

.50879 

.86089 

.52374 

.85188 

.53853 

.84261 

.55315 

.83308 

.56760 

.82330 

25 

36 

.50904 

.86074 

.52399 

.85173 

.53877 

.84245 

.55339 

.83292 

.56784 

.82314 

24 

37 

.50929 

.86059 

.52423 

.85157 

.53902 

.84230 

.65363 

.83276 

.5C808 

.82297 

23 

38 

.50954 

.86045 

.52448 

.85142 

.£3926 

.84214 

.55388 

.83260 

.56832 

.82281 

22 

39 

.50979 

.86030 

.52473 

.85127 

.63951 

.84198 

.55412 

.83244 

.56856 

.82264 

21 

40 

.51004 

.86015 

.52498 

.85113 

.53975 

.84182 

.55436 

.83228 

.56880 

.82248 

20 

41 

.51029 

.86000 

.52522 

.85096 

.54000 

.84167 

.55460 

.83212 

.56904 

.82231 

19 

42 

.51054 

.85985 

.52547 

.85081 

.54024 

.84151 

.55484 

.83195 

.56928 

.822141  18 

43 

.51079 

.85970 

.52572 

.85066 

.54049 

.84135 

.55509 

.83179  i  !  .56952 

.82198 

17 

44 

.51104 

.85956 

.52597 

.85051 

.54073 

.84120 

.55333 

.83163 

.66976 

.82181 

16 

45 

.51129 

.85941 

.52621 

.85035 

.54097 

.84104 

.55557 

.83147 

.67000 

.82165 

15 

46 

.51154 

.85926 

.52646 

.85020 

.54122 

.84088 

.55581 

.83131 

.57024 

.82148 

14 

47 

.51179 

.85911 

.52671 

.85005 

.54146 

.84072 

.55605 

.83115 

.67047 

.82132 

13 

48 

.51204 

.85896 

.52698 

.84989 

.54171 

.84057 

.55630 

.83098  .57071 

.82115 

12 

49 

.51229 

.85881 

.52720 

.84974 

.54195 

.84041 

.55654 

.82098 

11 

50 

.51254 

.85866 

.52745 

.84959 

.54220 

.84025 

.55678 

!83066 

.67119 

.83082 

10 

51 

.51279 

.85851 

.52770 

.84943 

.54244 

.84009 

.55702 

.83050 

.57143 

.82065 

9 

52 

.51304 

.85836 

.52794 

.84928 

.54209 

.83994 

.55726 

.83034 

.671G7 

.82048 

8 

53 

.51329 

.85821 

.52819 

.84913 

.54293 

.83978 

.55750 

.83017 

.57191 

.82032 

7 

54 

.51354 

.85806 

.52844 

.84897 

.54317 

.83962 

.55775 

.83001 

.67215 

.82015 

6 

55 

.51379 

.85792 

.52869 

.84882 

.54342 

.83946 

.55799 

.82985 

.67238 

.81999 

5 

56 

.51404 

.85777 

.52893 

.84866 

.54366 

.83930 

.55823 

.82969 

.67262 

.81982 

4 

57 

.51429 

.85762 

.52918 

.84851 

.54391 

.83915 

.55847 

.82953 

.67286 

.81965 

3 

58 

.51454 

.85747 

.52943 

.84836 

.54415 

.83899 

.55871 

.82936 

.57310 

.81949 

2 

59 

.51479 

.85732 

.52967 

.84820 

.54440 

.83883 

.65895 

.82920 

.67334 

.81932 

1 

60 

.51504 

.85717 

.52992 

.84805 

.54464 

.83867 

.55919 

.82904 

.57358 

.81915 

_0 

/ 

Cosin 

Sine 

Cosin 

Sine 

Cosin  Sine 

Cosin 

Sine 

Cosin 

Sine 

i 

59° 

58" 

57° 

56° 

55° 

NATURAL  SINES   AND   COSINES. 


119 


35C 


_Sine 

..'.rise 

.57381 
.57405 
.57429 
.57453 


.57501 
.57524 
.57548 
.57572 
.57596 

.57619 
.57643 
.57667 
.57691 
.57715 
.57738 
.57762 
.57786 
.57810 
.57833 

.57857 

.57881 


.57928 
.57952 
.57976 
.57999 


.58047 
.58070 

.58094 
.58118 
.58141 
.58165 


.58212 
.58236 
.58260 
58283 
.58307 


.58354 
.58378 
.58401 
.58425 
.58449 
.58472 
.58496 
.58519 
.58543 


.58590 
.58614 
.58637 
.58661 
.58684 
.58708 
.58731 
.58755 
^58779 

CoSH 


Cosin 

.81915 
.81899 
.81882 
.81865 
.81848 
.81832 
.81815 
.81798 
.81782 
.81765 
.81748 

.81731 

.81714 


,81681 


.81647 
.81631 
.81614 
.81597 
.81580 

.81563 
.81546 
.81530 
.81513 
.81496 
.81479 
.81463 
.81445 
.81423 
.81412 

.81395 

.81378 
.81201 
.81344 
.81327 
.81310 
.81293 
.81276 
.81259 
.81243 

.81225 
.81208 
.81191 
.81174 
.81157 
.81140 
.81123 
.81106 
.81089 
.81073 

.81055 
.81038 
.81021 
.81004 
.80987 
.80970 
.80953 


36° 


.80902 
Sine 


54- 


Sine 

.58779 


.58849 


.58920 
.58943 
.58967 
.58990 
.59014 

.59037 
.59061 
.59084 
.59108 
.59131 
.59154 
.59178 
.59201 
.59225 
.59248 

.59272 
.59295 
.59318 
.59342 
.59365 
.59389 
.59412 
.59436 
.59459 


.59529 
.59552 
.59576 
.59599 
.59022 
.59646 
.59G69 
.59G93 
.59716 

.59739 

.59763 
.50786 
.59809 


.59856 
.59879 
.59902 
.59926 
.59949 

.59972 


.60019 
.60042 
.60065 


.60112 
.60135 
.60158 
^60182 
Cosin 


Cosin 

.80902 
.80885| 
.80867! 
.808501 


80799 
80782 
80765 
80748 


80713 
80696 
80679 
80662 
80644 
80627 
,80610 


80576 
,80558 

80541 

,80524 


80472 
,80455 
,80438 
,804201 
,80403 


.80351 
.80334 
.80316 

.80299 
.80282 
.80264 
.80247 


.80212 

.80195 
.80178 
.80160 
.80143 
.80125 
.80108 


.80073 


.80021 

.80003 
.79986 
.799681 
.79951 | 
.79934 
.79916 
.79899 
.79881 
.79864! 
Sine  ! 


53° 


37° 


Sine 


.60205 
.60228 
.60251 
.60274 
.60298 
.60321 
.60344 
.60367 
.60390 
.60414 


.60506 
.60529 
.60553 
.60576 
.60599 
.60622 
.60645 


.60714 
.60738 
.60761 

.60784 


60853 
.60876 


.60922 
.60945 


.61015 
.61038 
.61061 
.61084 
.61107 

.61130 

.61153 
.61176 
.61199 
.61222 
.61245 


.61291 
.61314 
.61337 


.61383 
.61406 
.61429 
.61451 
.61474 
.61497 
.61520 
.61543 
.61566 


Cosin 


Cosin 

79864 
79846 
.79829 
.79811 
.79793 
79776 
.79758 
.79741 
.79723 
.79706 
.79688 

.79671 
.79653 
.79635 
.79618 
.79600 


.79565 
.79547 
.79530 
.79512 

.79494 
.79477 
.79459 
.79441 
.79424 
.79406 


79371 
79353 


.79318 
.79300 
.79282 
.79264 
.79247 
.79229 
.79211 
.79193 
.79176 
.79158 

.79140 

.79122 
.79105 
.79C87 
.79069 
.79051 
,79033 
,79016 
,78998 


.78944 


.78891 
.78873 
.78855 
.78837 
.78819 
.78801 
Sine 


88° 


52° 


Sine 
.61566 
.61589 
.61612 
.61635 
.61658 
.61681 
.61704 
.61726 
.61749 
.61772 
.61795 

.61818 
.61841 
.61864 
.61887 
.61909 
.619C2 
.61955 
.61978 


Cosin 

.78801 
.78783 
.78765 
.78747 
.78729 
.78711 
.78694 
.78676 
.78658 
.78640 


,62115 
.62138 


.62206 


.62274 
.62297 
.62320 
.62342 


.62411 


.62456 
.63479 


.62524 
.62547 
.62570 


.62615 
.62638 


.63706 


.62751 
.62774 
.62796 


Cosin 


78586 
78568 
78550 
78532 
78514 
78496 
78478 
78460 
78442 

78424 

78405 
78387 


,78351 
,78333 
78315 
,78297 
,78279 
,78261 

78243 


.78206 
.78188 
.78170 
.78152 
.78134 
.78116 
.78098 
.78079 

.78061 

.78043 
.78025 
.78007 
.77988 
.77970 
.77952 
.77934 
.77916 
.77897 

.77879 

.77861 
.77843 
.77824 
.77806 
.77788 
.77769 
.77751 
.77733 
77715 


Sine 


51' 


39° 


Sine 


.62977 
.63000 
.63022 
.63045 
.63068 
.63090 
.63113 
.63135 
.63158 

.63180 
.63203 
.63225 
.63248 
.63271 
.63293 
.63316 


.63383 

.63406 
.63428! 
.63451 | 
. 63473 ' 
.63496 
.635181 
. 63540 i 
.635631 
.63585 
.63608 

.63630 
.63653 
.63675 
.63698 
.63720 
.63742 
.63765 
.63787 
.63810 


.63877 
.63899 
.63922 
.63944 
.63966 
.63989 
.64011 
.64033 
.64056 

.64078 
.64100 
.64123 
.64145 
.64167 
.64190 
,64212 
,64234 
,64256 
,64279 


Cosin 


.77715 
.77696 
.77678 
.77660 
.77641 
.77623 
.77605 
.77586 
.77568 
.77550 
.77531 

.77513 

.77494 
.77476 
.77458 
.77439 
.77421 
.77402 
.77384 
.77366 
.77347 

.77329 
.77310 
.77292 
.77273 
.77255 
.77236 
.77218 
.77199 
.77181 
.77162 

.77144 
.77125 
.77107 
.77088 
.77070 
.77051 
.77033 
.77014 
.76996 
.76977 

.76959 

.76940 
.76921 
.76903 
.76884 
.76866 
.76847 
.76828 
.76810 
.76791 

.76772 
.76754 
.76735 
.76717 
.76698 
.76679 
.76661 
.76642 


Sine 


50° 


NATURAL  SINES  AND  COSINES. 


j 

40° 

41! 

42° 

43" 

44° 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine  Cosin 

Sine 

Cosin 

i 

~6 

.64279 

.76604 

.65606 

.75471 

.66913 

.74314 

.68200 

.73135 

.69466 

.71934  60 

l 

.64301 

.76586 

.65628 

.75452 

.66935 

.74295 

.68221 

.73116 

.69487 

.71914  59 

2 

.64323 

.76567 

.65650 

.75433 

.66956 

.74276 

.68242 

.73096 

.69508 

.71894  58 

3 

.64346 

.70548 

.65672 

.75414 

.66978 

.74256 

.68264 

.73076 

.69529 

.71873  57 

4 

.64308 

.76530 

.65694 

.75395 

.66999 

.74237 

.68285 

.73056 

.69549 

.71853  56 

5 

.64393 

.70511 

.65716 

.75375 

.67021 

.74217 

.68306 

.73036 

.69570 

.fl8C3  55 

6 

.64412 

.70492 

.65738 

.75356 

.67045 

.74198 

.68327 

.73016 

.69591 

.71813  54 

7 

.64435 

.70473 

.65759 

.75337 

.67064 

.74178 

.68349 

.72996 

.69612 

.71792  53 

8 

.64457 

.76455 

.65781 

.75318 

.67086 

.74159 

.68370 

.72976 

.69033  j.  71772 

52 

9 

.64479 

.76433 

.65803 

.75299 

.67107 

.74139 

.68391 

.72957 

.69654s.  71752 

51  ' 

10 

.64501 

.76417 

.65825 

.75280 

.67129 

.74120 

.68412 

.72937 

.69675 

.71752 

50  , 

11 

.64524 

.76393 

.65847 

.75261 

.67151 

.74100 

.68434 

.72917 

.69696 

.71711 

49 

12 

.64546 

.70333 

.65839 

.75241 

.67172 

.74080 

.68455 

.72807 

.69717 

.71691 

48 

13 

.64508 

.76361 

.65891 

.75222 

.67194 

.74061 

.68476 

.72877 

.69737 

.71671 

47 

14 

.64590 

.76342 

.65913 

.75203 

.67215 

.74041 

.68497 

.72857 

.69758 

.71650 

46 

15 

.64612 

.76323 

.65935 

.75184 

.67237 

.74022 

.68518 

.728C7 

.69779 

.71630 

43 

16 

.64635 

.76304 

.65956 

.75165 

.67258 

.74003 

.68539 

.72817 

.69800 

.71610 

41 

17 

.64657 

.70236, 

.65378 

.75146 

.67280 

.73983 

.68561 

.727C7 

.69821 

.71590 

43 

18 

.64679 

.76287 

.G3000 

.75126 

.67301 

.73903 

.68582 

72777 

.69842 

.71569 

42 

19 

.64701 

.762431 

.63023 

.75107 

.67323 

.73944 

.68603 

(^o1*""^ 

.69CC21.  71549 

41 

20 

.64723 

.70229 

.63044 

.75088 

.67344 

.73924 

.68624 

!  72737 

.69883 

.71529 

40 

21 

.64746 

.76210 

.66066 

.75069 

.67366 

.73904 

.68645 

.72717 

.69904 

.71508 

39 

22 

.64703 

.73192 

.63033 

75050 

.  G7C87 

.733C3 

.63CC3 

.72637 

.69925  .71488 

33 

23 

.64790 

.73173i 

.63103 

75033  I 

.67409 

.73803 

.68083 

.72677 

.699461.71468 

37 

<vl 

.64812 

.73154  ! 

.63131 

75011  1 

.67430 

73843 

.687C9 

.72657 

.69966 

.71447 

36 

25 

.64834 

.76135 

.63153 

74933 

.67452 

73823 

.687C3 

.72637 

.69987 

.71427 

35 

26 

.64856 

.76116 

.63175 

74973 

.67473 

73803 

.68751 

.72617 

.70008 

.71407 

84 

27 

.64878 

.76097 

.65197 

74953 

.67495 

73787 

.68772 

.72597 

.70029 

.71386 

33 

28 

.64901 

.76078 

.63218 

74334 

67516 

73767 

.68793 

.72577 

.70049 

.71360 

32 

29 

.64923 

.76059 

.68240 

74315 

67533 

73747 

.63314 

.72557 

.70070 

.71345 

31 

30 

.64945 

.76041 

.63262 

74896 

67559 

73728 

.68835 

.  72537  . 

.70091 

.71325 

30 

31 

.64967 

.76022 

.66284 

74376 

67580 

73708 

.68857 

.72517 

.70112 

.71305 

29 

32 

.64989 

.76003 

.63303 

743571 

67632 

73683 

.68078 

.72497 

70132 

.71284 

28 

33 

.65011 

.75984 

.63327 

74833 

67623 

73663 

.68803 

.72477 

.70153 

.71204 

27 

34 

.65033 

.75965 

.60349 

74318 

G7G45 

73643 

.68920 

.724571 

.70174 

.71243 

26 

35 

.65055 

.7594'* 

.63371 

74793 

67G33 

73023 

.63341 

.72437 

.70195 

.71223 

35 

36 

.65077 

.75927 

.63393 

74783 

67038 

73010 

.63963 

.72417 

.7C215 

.71203 

24 

37 

.65100 

.75908 

.66414 

74700 

67709 

73590 

.68983 

72397 

.70236  .71183 

23 

38 

.65122 

.75889 

.66436 

74741 

67730 

73570 

.69004 

72377 

.70257 

.71162 

22 

39 

.65144 

.75870 

.63433 

74722 

67752 

73551 

.69025 

72357 

.70277 

.71141 

21 

40 

.65166 

.75851 

.66480 

74703 

67773 

73531 

.69046 

72337 

.70298 

.71121 

20 

41 

.65188 

.75832 

.66501 

74683 

67795 

73511 

.69067 

72317 

.70319 

.71100 

19 

42 

.65210 

.75813 

.603^3 

74634 

67816 

73401 

.63033 

72297 

.70339 

.71080 

13 

43 

.65232 

.75794 

.66543 

74644 

67837 

73472 

.69103 

72277 

.70360 

.71059 

17  1 

44 

.65254 

.75775 

.66503 

74625 

67859 

73432 

.63130 

72257 

.70381 

.71039 

10 

45 

.65276 

.75756 

.66583 

7460S 

67880 

73432 

.69151 

72236 

.70401 

.71019 

15 

46 

.65298 

.75738 

.63610 

74533 

67901 

73413 

.69172 

72216 

.70422 

.70998 

14 

47 

.65320 

.75719 

.66632 

74567  j 

67923 

73393 

.69193 

72196 

.70443 

.70978 

13 

43 

.65342 

.75700 

.66653 

74548, 

67944 

73373 

.69214 

72176 

.70463 

.70957 

12 

43 

.65364 

.75680 

.68675 

74528 

.679G5 

73353 

.69235 

72156 

.70484 

.70937 

11 

50 

.65386 

.75661 

.66697 

74509 

67987 

73333 

.69256 

72136 

.70505 

.70916 

10 

51 

.65408 

.75642 

.66718 

74489 

.68008 

73314 

.69277 

72116 

.70525 

.70896 

9 

53 

.65430 

.75623 

.66740 

74470! 

.68029 

73294 

.69298 

72095 

.70546 

.70875 

8 

53 

.65452 

.75604 

.66763 

.74451 

.68051 

73274 

.69319 

72075 

.70567 

.70855 

r* 

54 

.65474 

.75585 

.66783 

.74431 

.68072 

73254 

.69340 

72055 

.70587 

.70834 

6 

55 

.65496 

.75566 

.66805 

.74412 

.68093 

73234 

.69301 

72035 

.70608 

.70813 

5 

56 

.65518 

.75547 

.66827 

.74392 

.68115 

73215 

.69382 

72015 

.70628 

.70793 

4 

57 

.65540 

.75528 

.668481.74373 

.68136 

73195 

.69403 

71995 

.70649 

.70772 

3 

:3 

.65562 

.75509: 

.66870 

.74353 

.68157 

73175 

.69424 

71974 

.70670 

.70752 

2  ! 

59 

.65584 

.75490 

.66891 

.74334 

.68179 

73155 

.69445 

71954 

.70690 

.70731 

1  ! 

GO 

.65606 

.75471 

.66913 

.74314 

.68200 

73135 

.69466 

71934 

.70711 

.70711 

a 

t 

Cosiu 

Sine 

Cosin 

Sine  | 

Cosin 

Sine 

Cosin  |  Sine 

Dosin 

Sine 

r 

49° 

48° 

47° 

46°   1 

45° 

\ 

NATURAL  TANGENTS  AND  COTANGENTS. 


0° 

1-           !!           2° 

8° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tan?:     Cotang 

0 

.00000 

Infinite. 

.01746 

57.2900 

.03492 

28.6363 

.05241 

19.0811 

60 

1 

.00029 

3437.75 

.01775 

56.3506 

.03521 

28.3994 

.05270 

18.9755 

50 

2 

.00058 

1718.87 

.01804 

55.4415 

.03550 

28.16G4 

.05299 

18.8711 

58 

3 

.00087 

1145.92 

.01833 

54.5613 

.03579 

27.9372 

.05328 

18.7678 

57 

4 

.00116 

859.436 

.01862 

53.7086 

.03609 

27.7117 

.05357 

18.6656 

5(5 

5 

.00145 

637.549 

.01891 

52.8821 

.03G38 

27.4099 

.05387 

18.5645 

65 

6 

.00175 

572.957 

.01920 

52.0807 

.03GG7 

27.2715 

.05416 

18.4G45 

51 

7 

.00204 

491.106 

.01949 

51.3032 

.03C06 

27.05G6 

.05445 

18.3G55 

53 

8 

.00233 

429.718 

.01978 

50.5485 

.03725 

26.8450 

.05474 

18.2G77    E2 

9 

.00262 

831.971 

.02007 

49.8157 

.03754 

26.63G7 

.05503 

18.1708    51  i 

10 

.00291 

343.774 

.02036 

49.1039 

.03783 

26.4316 

.05533 

18.0750 

50 

11 

.00320 

312.521 

.02066 

48.4121 

.03812 

26.2296 

.05562 

17.9802 

49 

12 

.00349 

236.478 

.02095 

47.7395 

.03842 

26.0307 

.05591 

17.88G3 

48 

13 

.00378 

234.441 

.02124 

47.0353 

.03871 

25.8348 

.05620 

17.7934 

47 

1  \ 

.00407 

245.552 

.02153 

46.4489 

.03900 

25.6418 

.05649 

17.7015 

46 

15 

.00436 

229.182 

.02182 

45.8294 

.03929 

25.4517 

.05678 

17.6106 

45 

16 

.00465 

214.858 

.02211 

45.2261 

.03958 

25.2644 

.05708 

17.5205 

44 

17 

.00495 

202.219 

.02240 

44.6386 

.03987 

25.0798 

.05737 

17.4314 

43 

18 

.00524 

190.984 

.02269 

44.0661 

.04016 

24.8978 

.05766 

17.3432 

42 

10 

00553 

180.932 

.02298 

43.5081 

.04046 

24.7185 

.05795 

17.2558 

41 

20 

00583 

171.885 

.02328 

42.9641 

.04075 

24.5418 

.05824 

17.1693 

40 

21 

00611 

163.700 

.02357 

42.4335 

.04104 

24.3675 

.05854 

17.0837 

39 

.-  > 

OOG-10 

156.259 

.C2386 

41.9158 

.04133 

24.1957 

.05G33 

16.9990 

38 

23 

OOCC3 

119.405 

.C2415 

41.4106 

.04162 

24.02G3 

.05012 

16.9150 

37 

24 

OOGC3 

113.237 

.02444 

40.9174 

.04191 

23.8593 

.05041 

16.8319 

36 

25 

.00727 

137.507 

.02473 

40.4358 

.04220 

23.6945 

.05970 

16.7496 

35 

,-:;; 

.00756 

132.219 

.02503 

39.9655 

.04250 

23.5321 

.05999 

16.6G81 

34 

•27 

.007G5 

327.321 

.02531 

39.5059 

.04279 

23.3718 

.06029 

16.5874 

33 

2S 

.CC315 

122.774 

.02560 

39.0568 

.04308 

23.2137 

.OGC58 

16.5075 

32 

39 

.CC344 

118.540 

.02589 

38.6177 

.04337 

23.0577 

.06087 

16.4283 

31 

SO 

.00873 

114.689 

.02619 

38.1885 

.04366 

22.9038 

.06116 

16.3499 

30 

31 

.00902 

110.892 

.02648 

37.7686 

.04395 

22.7519 

.06145 

16.2722 

29 

.00931 

107.426 

.02077 

37.3579 

.04424 

22.6020 

.06175 

16.1952 

28 

33 

.009CO 

104.171 

.02706 

36.9560 

.04454 

22.4541 

.06204 

16.1190 

27 

84 

.009G9 

101.107 

.02735 

36.5627 

.04483 

22.3081 

.06233 

16.0435 

2G 

35 

.01018 

£3.2179 

.02764 

36.1776 

.04512 

22.1640 

.062C2 

15.9687 

25 

36 

.01047 

95.4895 

.02793 

35.8006 

.04541 

22.0217 

.06291 

15.8945 

21 

::7 

.01076 

92.9085 

.02822 

35.4313 

.04570 

21.8813 

.06321 

15.8211 

23 

38 

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90.4633 

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35.0695 

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21.7426 

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15.7483 

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88.1436 

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34.7151 

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21.6056 

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15.6762 

21 

40 

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85.9398 

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34.3678 

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21.4704 

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15.6048 

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83.8435 

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34.0273 

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15.5340 

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81.8470 

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33.6935 

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21.2049 

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15.4638 

18 

13 

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79.9434 

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33.3662 

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21.0747 

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15.3943 

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11 

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78.1263 

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33.0452 

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20.9460 

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15.3254 

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76.3900 

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32.7303 

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20.8188 

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15.2571 

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32.4213 

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73.1390 

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32.1181 

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60.3058 

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29.3711 

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59.2659 

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29.1820 

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19.2959 

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Cotang 

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Cotang 

Tang 

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89a 

88"          li           87* 

06° 

122 


NATURAL  TANGENTS  AND  COTANGENTS 


4° 

5°              1 

1            6°            :[            7° 

Tang 

Cotang 

Tang 

Cotang 

_Tang_ 

Cotang 

Tang 

Cotang 

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.06993 

14.3007 

.08749 

11.4301 

.10510 

9.51436 

.12278 

8.14435 

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14.2411 

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11.3919 

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9.48781 

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8.12481 

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2 

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14.1821 

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11.3540 

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9.46141 

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8.195SG 

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14.1235 

.08837 

11.3163 

.10599 

9.43515 

.12367 

8.C8600 

.07110 

14.0655 

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11.2789 

.10628 

9.40904 

.12397 

8.C6C74 

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14.0079 

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11.2417 

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9.38307 

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8.04756 

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11  .2048 

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9.35724 

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8.C204S 

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13.8940 

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11.1681 

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9.33155 

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8.00948 

63 

8 

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13.8378 

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11.1316 

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9.30599 

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7.9C058 

62 

9j   .07256 

13.7821 

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11.0954 

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9.28058 

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7.97176 

61 

10 

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13.7267 

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11.0594 

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9.25530 

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7.95302 

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11 

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13.6719 

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11.0237 

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9.23016 

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7.93438 

40 

12 

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13.6174 

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10.9882 

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9.20516 

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7.91582 

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13.5634 

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10.9529 

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9.18028 

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7.89734 

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13.5098 

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9.15554 

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15 

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13.4566 

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10.8829 

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9.13093 

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7.86064 

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16 

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13.4039 

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10.8483 

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9.10646 

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7.84242 

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17 

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13.3515 

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10.8139 

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13.2996 

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10.7797 

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9.057C9 

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7.80622 

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10 

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13.2480 

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10.7457 

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9.03379 

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7.78825 

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13.1969 

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10.7119 

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9.00983 

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7.77035 

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21 

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13.1461 

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10.6783 

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8.98598 

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7.75254 

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22 

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13.0958 

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10.6450 

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8.96227 

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7.72480 

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23 

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13.0458 

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10.6118 

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8.93867 

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7.71715 

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12.9962 

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10.5789 

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8.91520 

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25 

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12.9469 

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10.5462 

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8.89185 

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7.68208 

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12.8981 

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10.5136 

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8.86862 

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7.66466 

34 

27 

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10.4813 

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8.84551 

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7.64732 

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28 

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7.63005 

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29 

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12.7536 

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10.4172 

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8.79964 

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30 

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12.7062 

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10.3854 

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8.77689 

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7.59575 

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31 

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12.6591 

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10.3538 

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8.75425 

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7.57872 

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32 

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12.6124 

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10.3224 

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8.73172 

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12.5660 

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10.2913 

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8.70931 

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7.54487 

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34 

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10.2602 

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8.68701 

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7.52806 

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12.4742 

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10.2294 

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8.66482 

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7.51132 

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36 

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12.4288 

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10.1988 

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8.64275 

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7.49465 

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37 

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12.3838 

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10.1683 

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8.62078 

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7.47806 

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12.3390 

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10.1381 

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8.59893 

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7.46154 

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12.2946 

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10.1080 

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8.57718 

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7.44509 

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40 

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12.2505 

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10.0780 

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8.55555 

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7.42871 

20 

41 

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12.2067 

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10.0483 

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8.53402 

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7.41240 

19 

42 

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12.1632 

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10.0187 

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8.51259 

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7.39616 

18 

43 

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12.1201 

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9.98931 

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8.49128 

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7.37999 

17 

44 

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12.0772 

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9.96007 

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8.47007 

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7.3G889    16 

45 

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12.0346 

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9.93101 

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8.44896 

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7.34786 

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46 

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11.9923 

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9.90211 

1   .11865 

8.42795 

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7.33190 

14 

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11.9504 

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9.87338 

|   .11895 

8.40705 

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7.31600 

13 

48 

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11.9087 

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9.84482 

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8.38625    1    .13698 

7.20018 

12 

49 

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11.8673 

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9.81641 

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8.36555 

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7.  £8442 

11 

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11.8262 

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9.78817 

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8.34496 

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7.26873 

10 

61 

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11.7853 

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8.76009 

1  .12013 

8.32446 

i   .13787 

7.25310 

9 

52 

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11.7448 

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9.73217 

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8.30406 

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7.23754 

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53 

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11.7045 

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9.70441 

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8.28276 

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7.22204 

7 

54 

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11.6645 

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9.67680 

8.26355 

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7.20661 

6 

55 

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11.6248 

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9.64935 

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8.24345 

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7.19125 

6 

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11.5853 

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8.22344 

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7.17594 

4 

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11.5461 

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9.59490 

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8.20G52 

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7.16071 

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11.5072 

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9.56791 

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8.18370 

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7.14553 

g 

£3 

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11.4685 

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9.54106 

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8.16398 

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11.4301 

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9.51436 

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8.14435 

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7.11537 

0 

t 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

9 

85' 

:       84° 

83* 

82* 

NATURAL   TANGENTS   AND   COTANGENTS. 


123 


8°            I!            9° 

10° 

IP 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.14054 

7.11537 

.15838 

6.31375 

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5.67128 

.19438 

5.14455 

80 

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7.10038 

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6.30189 

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5.66165 

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5.13658 

59 

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7.08546 

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6.29007 

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5.65205 

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5.12862 

68 

3 

.14143 

7.07059 

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6.27829 

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5.64248 

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5.12069 

57 

4 

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7.05579 

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6.26655 

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5.63295 

.19559 

5.11279 

56 

j 

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7.04105 

.15988 

6.25486 

.17783 

5.62344 

.19589 

5.10490 

55 

6 

.14232 

7.02637 

.16017 

6.24321 

.17813 

5.61397 

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5.09704 

54 

7 

.14262 

6.91174 

.16047 

6.23160 

.17843 

5.60452 

.19649 

5.08921 

53 

8 

.14291 

6.99718 

.16077 

6.22003 

.17873 

5.59511 

.19680 

5.08139 

52 

9 

.14321 

6.98268 

.16107 

6.20851 

.17903 

5.58573 

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5.07360 

51 

10 

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6.96823 

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6.19703 

.17933 

5.57638 

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5.06584 

50 

11 

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6.95385 

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6.18559 

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5.56706 

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5.05809 

49 

12 

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6.93952 

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6.17419 

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5.55777 

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5.05037 

48 

13 

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6.92525 

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6.16283 

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5.54851 

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5.04267 

47 

14 

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6.91104 

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6.15151 

.18053 

5.53927 

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5.03499 

46 

15 

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6.89688 

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6.14023 

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5.53007 

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5.02734 

45 

16 

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6.88278 

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6.12899 

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5.52090 

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5.01971 

44 

17 

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6.86874 

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6.11779 

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5.51176 

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5.01210 

43 

18 

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6.85475 

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6.10664 

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5.50264 

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5.00451 

42 

19 

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6.84082 

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6.09552 

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5.49356 

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4.99695 

41 

20 

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6.82694 

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6.08444 

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5.48451 

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4.98940 

40 

21 

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6.81312 

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6.07340 

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5.47548 

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4.98188 

39 

22 

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6.79936 

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6.06240 

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5.46648 

.20103 

4.97438 

38 

23 

.14737 

6.78564 

.16525 

6.05143 

.18323 

5.45751 

.20133 

4.96690 

37 

24 

.14767 

6.77199 

.16555 

6.04051 

.18353 

5.44857 

.20164 

4.95945 

36 

25 

.14796 

6.75838 

.16585 

6.02962 

.18384 

5.43966 

.20194 

4.95201 

35 

25 

.14826 

6.74483 

.16615 

6.01878 

.18414 

5.43077 

.20224 

4.94460 

34 

27 

.14856 

6.73133 

.16645 

6.00797 

.18444 

5.42192 

.20254 

4.93721 

33 

23 

.14886 

6.71789 

.16674 

5.99720 

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5.41309 

.20285 

4.92984 

32 

£3 

.14915 

6.70450 

.16704 

5.98646 

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5.40429 

.20315 

4.92249 

31 

RO 

.14945 

6.69116 

.16734 

5.97576 

.18534 

5.89552 

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4.91516 

30 

81 

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6.67787 

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5.96510 

.18564 

5.38677 

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4.90785 

2$ 

3-2 

.15005 

6.66463 

.16794 

5.95448 

.18594 

5.37805 

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4.90056 

28 

C3 

.15034 

6.65144 

.16824 

5.94390 

.18624 

5.36936 

.20436 

4.89330 

27 

34 

.15064 

6.63831 

.16854 

5.93365 

.18654 

5.36070 

.20466 

4.88605 

26 

35 

.15094 

6.62523 

.16834 

5.92283 

.18684 

5.35206 

.20497 

4.87882 

25 

3G 

.15124 

6.61219 

.16914 

5.91236 

.18714 

5.34345 

.20527 

4.87162 

24 

37 

.15153 

6.59921 

.16944 

5.90191 

.18745 

5.33487 

.20557 

4.86444 

23 

C3 

.15183 

6.53627 

.16974 

5.89151 

.18775 

5.32631 

.20588 

4.85727 

22 

C9 

.15213 

6.57339 

.17004 

5.88114 

.18805 

5.31778 

.20618 

4.85013 

21 

40 

.15243 

6.56055 

.17033 

5.87080 

.18835 

5.30928 

.20648 

4.84300 

20 

41 

.15272 

6.54777 

.17063 

5.86051 

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5.30080 

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4.83590 

19 

42 

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6.53503 

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5.85024 

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5.29235 

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4.82882 

18 

43 

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6.52234 

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5.84001 

.18925 

5.28393 

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4.82175 

17 

44 

.15362 

6.50970 

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5.82982 

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5.27553 

.20770 

4.81471 

16 

45 

.15391 

6.49710 

.17185 

5.81966 

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5.26715 

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4.80769 

15 

46 

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6.48456 

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5.80953 

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5.25880 

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4.80068 

14 

47 

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6.47206 

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5.79944 

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5.25048 

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4.79370 

13 

48 

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6.45961 

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5.78938 

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5.24218 

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4.78673 

12 

49 

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6.44720 

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5.77936 

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5.23391 

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4.77978 

11 

50 

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6.43484 

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5.76937 

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5.22566 

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4.77286 

10 

51 

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6.42253 

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5.75941 

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5.21744 

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4.76595 

9 

52 

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6.41026 

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5.74949 

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5.20925 

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4.75906 

8 

53 

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6.39804 

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5.73960 

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5.20107 

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4.75219 

7 

54 

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6.38587 

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5.72974 

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5.19293 

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4.74534 

6 

55 

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6.37374 

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5.71992 

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5.18480 

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4.73851 

r 

50 

.15719 

6.36165 

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5.71013 

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5.17671 

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4.73170 

4 

57 

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6.34961 

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5.70037 

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5.16863 

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4.72490 

8 

53 

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6.33761 

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5.69064 

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5.16058 

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4.71813 

2 

53 

.15809 

6.32566 

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5.68094 

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5.15256 

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4.71137 

1 

60 

.15838 

6.31375 

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5.67128 

.19438 

5.14455 

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4.70463 

0 

i 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

81° 

80" 

79° 

78° 

NATURAL  TANGENTS   AND   COTANGENT?, 


12° 

13° 

14° 

15° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang  |  Cotang 

0 

.21256 

4.70463 

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4.33148 

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4.01078 

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3.73205 

60 

1 

.21286 

4.69791 

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4.32573 

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4.00582 

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3.72771 

59 

2 

.21316 

4.69121 

.23148 

4.32001 

.24995 

4.00086 

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3.72a38 

58 

3 

.21347 

4.C8452 

.23179 

4.31430 

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3.99592 

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3.71907 

57 

4 

.21377 

4.67786 

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4.30860 

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3.99099 

.26920 

3.71476 

56 

5 

.21408 

4.67121 

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4.30291 

.25087 

3.98607 

.26951 

3.71046 

55 

G 

.21438 

4.66458 

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4.29724 

.25118 

3.98117 

.26982 

3.70G16 

54 

7 

.21469 

4.65797 

.23301 

4.29159 

.25149 

3.97627 

.27013 

3.70188 

53 

8 

.21499 

4.65138 

.23332 

4.28595 

.25180 

3.97139 

.27044 

3.697G1 

52 

9 

.215£9 

4.64480 

.23363 

4.28032 

.25211 

3.96651 

.27076 

3.69335 

51 

10 

.21560 

4.63825 

.23393 

4.27471 

.25242 

3.96165 

.27107 

3.68909 

50 

11 

.21590 

4.63171 

.23424 

4.26911 

.25273 

3.95680 

.27138 

3.68485 

49 

12 

.21621 

4.62518 

.23455 

4.26352 

.25304 

3.95196 

.271G9 

3.680G1 

48 

13 

.21051 

4.61868 

.23485 

4.25795 

.25335 

3.94713 

.27201 

3.67638 

47 

14 

.21682 

4.61219 

.23516 

4.25239 

.25366 

3.94232 

.27232 

3.67217 

46 

15 

.21712 

4.60572 

.23547 

4.24685 

.25397 

3.93751 

.27263 

3.6G796 

45 

16 

.21743 

4.59927 

.23578 

4.24132 

.25428 

3.93271 

.27294 

3.66376 

44 

17 

.21773 

4.59283 

.23608 

4.23580 

.25459 

3.92793 

.27326 

3.65957 

43 

18 

.21804 

4.58G41 

.23639 

4.23030 

.25490 

3.92316 

.27357 

3.65538 

42 

19 

.21834 

4.58001 

.23670 

4.22481 

.25521 

3.91839 

.27388 

3.65121 

41 

20 

.21864 

4.57363 

.23700 

4.21933 

.25552 

3.91364 

.27419 

3.64705 

40 

21 

.21895 

4.56726 

.23731 

4.21387 

.25583 

3.90890 

.27451 

3.64289 

39 

01 

.21025 

4.56091 

.23762 

4.20842 

.25G14 

3.90417 

.27482 

3.63874 

38 

23 

.21956 

4.55458 

.23793 

4.20298 

.25G45 

3.89945 

.27513 

3.63461 

37 

24 

.21986 

4.54826 

.23823 

4.19756 

.25G76 

3.89474 

.27545 

3.63048 

36 

25 

.22017 

4.54196 

.23854 

4.19215 

.25707 

3.89004 

.27576 

3.62G36 

}5 

2G 

.22047 

4.53568 

.23885 

4.18675 

.25738 

3.88536 

.27607 

3  62224 

34 

27 

.22078 

4.52941 

.23916 

4.18137 

.25769 

3.880G8 

.27638 

3.61814 

33 

28 

.22108 

4.52316 

.23946 

4.17600 

.25800 

3.87601 

.27670 

3.61405 

32 

29 

.22139 

4.51693 

.23977 

4.17064 

.25831 

3.87136 

.27701 

3.60996 

31 

30 

.22169 

4.51071 

.24008 

4.16530 

.25862 

3.86671 

.27732 

360588 

30 

31 

.22200 

4.50451 

.24039 

4.15997 

.25893 

3.86208 

.27764 

3.60181 

29 

32 

.22231 

4.49832 

.24069 

4.15465 

.25924 

3.85745 

.27795 

3.59775 

28 

33 

.22261 

4.49215 

.24100 

4.14934 

.25955 

3.85284 

.27826 

3.59370 

27 

34 

.22293 

4.48600 

.24131 

4.14405 

.25986 

3.84824 

.27858 

3.58966 

26 

35 

.22322 

4.47986 

.24162 

4.13877  1 

.26017 

3.84364 

.27889 

3.585G2 

25 

36 

.22353 

4.47374 

.24193 

4.13350 

.26048 

3.83906 

.27921 

3.58160 

24 

37 

.22383 

4.46764 

.24223 

4.12825 

.26079 

3.83449 

.27952 

3.57758 

23 

38 

.22414 

4.46155 

.24254 

4.12301 

.26110 

3.82992 

.279&3 

3.57357 

22 

39 

.22444 

4.45548 

.24285 

4.11778 

.26141 

3.82537 

.28015 

3.56957 

21 

40 

.22475 

4.44942 

.24316 

4.11256 

.26172 

3.82083 

.28046 

3.56557 

20 

41 

.22505 

4.44338 

.24347 

4.10736 

.26203 

3.81630 

.28077 

3.56159 

19 

42 

.22536 

4.43735 

.24377 

4.10216 

.26235 

3.81177 

.28109 

3.55761 

18 

43 

.22567 

4.43134 

.24408 

4.09699 

.26266 

3.80726 

.28140 

3.55364 

17 

44 

.22597 

4.42534 

.24439 

4.09182 

.26297 

3.80276 

.28172 

3.54968 

16 

45 

.22628 

4.41936 

.24470 

4.08666 

.26328 

3.79827 

.28203 

3.54573 

15 

4G 

.22658 

4.41340 

.24501 

4.08152 

.26359 

3.79378 

.28234 

3.54179 

14 

47 

.22G89 

4.40745 

.24532 

4.07639 

.26390 

3.78931 

.28266 

3.53785 

13 

48 

.22719 

4.40152 

.24562 

4.07127 

.26421 

3.78485 

.28297 

8.53393 

12 

49 

.22750 

4.39560 

.24593 

4.06616 

.26452 

3.78040 

.28329 

3.53001 

11 

50 

.22781 

4.38969 

.24624 

4.06107 

.26483 

3.77595 

.28360 

3.52609 

10 

51 

.22811 

4.38381 

.24655 

4.05599 

.26515 

3.77152 

.28391 

3.52219 

9 

52 

.22842 

4.37793 

.24686 

4.05092 

.26546 

3.76709 

.28423 

3.51829 

8 

53 

.22872 

4.37207 

.24717 

4.04586 

.26577 

3.76268 

.28454 

3.51441 

7 

54 

.22903 

4.36G23 

.24747 

4.04081 

.26608 

3.75828 

.28486 

3.51053 

6 

55 

.22934 

4.3G040 

.24778 

4.03578 

.26639 

3.75388 

.28517 

3.50666 

5 

58 

.22964 

4.35459 

.24809 

4.03076 

.26670 

3.74950 

.28549 

3.50279 

4 

57 

.22995 

4.34879 

.24840 

4.02574 

.26701 

3.74512 

.28580 

3.49894 

3 

58 

.23026 

4.34300 

.24871 

4.02074 

.26733 

3.74075 

.28612 

3.49509 

2 

59 

.23056 

4.33723 

.24902 

4.01576 

.26764 

3.73640 

.28643 

3.49125 

1 

CO 

.23087 

4.33148 

.24933 

4.01078 

.26795 

3.73205 

.28675 

3.48741 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

77°            ,           76°           ll           75°  •     ,||           74» 

NATURAL  TANGENTS   AND   COTANGENTS. 


125 


16° 


Tang 

.28675 
.28706 


.28800 


.28864 


.28958 


.29053 

.29084 


.29147 
.29179 
.29210 
.29242 
.29274 


49  .30224 
50 


.29400 
.29432 
.29463 
.29495 
.29526 
.29558 
.29590 
.29621 

.29653 


.29716 
.29748 


.29811 
.29843 
.29875 
.29906 


.29970 
.30001 
.30033 
.30065 
.30097 
.30128 


.30287 
.30319 
.30351 


.30414 
.30446 
.30478 


.30541 

.30573 

Cotang 


Cotang 


3.48741 
3.4&S59 
3.47977 
3.47596 
3.47216 


3.46080 
3  45703 
3.45327 
3.44951 

3.44576 
3.44202 
3.43829 
3.43456 
3.43084 
3.42713 
3.42343 
3.41973 
3.41604 
3.41236 

3.40869 
3.40502 
3.40136 
3.39771 
3.39406 
3.39042 
3.38679 
3.38317 
3.37955 
3.37594 

3.37234 

3.36875 
3.36516 
3.36158 
3.35800 
3.35443 
3.3508? 
3.34732 
3.34377 
3.34023 


3.33317 
3.32965 
3.32614 
3.32264 
3.31914 
3.31565 
3.31216 


3.30521 
3.30174 


3. 

3.29139 

3.28795 

3.28452 

3.28109 

3.27767 

3.27426 

3.27085 


Tang 


.30573 
.30605 
.30637 


.30700 
.30732 
.30764 
.30796 


.30923 


.30987 
.31019 
.31051 
.31083 
.31115 
.31147 
.31178 
.31210 


.31274 
.31306 


.31370 
.31402 
.31434 
.31466 
.31498 
.31530 

.31562 
.31594 


.31658 
.31690 
.31722 
.31754 
.31786 
.31818 
.31850 

.31882 
.31914 
.31946 
.31978 
.32010 
.32042 
.32074 
.32106 
.32139 
.32171 


.32235 


.32331 


.32428 
.32460 


Cotang 


Cotang 


3.27085 
3.26745 
3.26406 
3.26067 
3.25729 


3.25055 
3.24719 
3.24383 
3.24049 
3.23714 


3.23048 
3.22715 


3.22053 
3.21722 
3.21392 
3.21063 
3.20734 
3.20406 

3.20079 
3.19752 
3.19426 
3.19100 
3.18775 
3.18451 
3.18127 
3.17804, 
8.17481 
3.17159 

3.16838 
3.16517 
3.16197 
3.15877 
3.15558 
3.15240 
3.14922 
3.14605 
3.14288 
3.13972 

3.13656 
3.13341 
3.13027 
3.12713 
3.12400 
3.12087 
3.11775 
3.11464 
3.11153 
3.10843 

3.10532 
3.10223 
3.09914 
3.09606 
3.09298 
3.08991 


3.08379 
3.08073 
3.07768 


Tang 


18° 


Tang 


.32524 
.32556 


.32621 


.32717 
.32749 

.32782 
.32814 


72a 


.32911 


.32975 


.33040 
.33072 
.33104 


.33201 


.33330 


.33427 
.33460 


.33492 


.33557 
.33589 


.33718 
.33751 
.33783 

.33816 


.33945 


.34010 
.34043 
.34075 
.34108 

.34140 
.34173 
.34205 


.34270 
.34303 
.34335 
.34368 
.34400 
.34433 
Cotang 


Cotang 

3.07768 
3.07464 
3.07160 
3.06857 
3.06554 
3.06252 
3.05950 
3.05649 
3.05349 
3.05049 
3.04749 

3.04450 
3.04152 


3.03556 


3.02963 
3.02667 
3.02372 
3.02077 
3.01783 


3.01196 
3.00903 
3.00611 
3.00319 
3.00028 
2.99738 
2.99447 
2.99158 


2.98292 
2.98004 
2.97717 
2.97430 
2.97144 
2.96858 
2.96573 
2.96288 
2.96004 

2.95721 
2.95437 
2.95155 
2.91872 
2.94591 


2.94028 
2.93748 
2.934G8 
2.93189 

2.92910 
2.92632 
2.92354 
2.92076 
2.91799 
2.91523 
2.91246 
2.90971 
2.90696 
2.90421 


Tang 


19° 


Tang 
.34433 
.34465 
.34498 
.34530 
.34563 
.34596 


.34661 
.34693 
.34726 
.34758 

.34791 


.34856 


.34922 
.34954 
.34987 
.35020 
.35052 
.35085 

.35118 
.35150 
.35183 
.35216 
.35248 
.35281 
.35314 
.35346 
.35379 
.35412 

.35445 
.35477 
.35510 
.35543 
.35576 
.35608 
.35641 
.35674 
.35707 
.35740 

.35772 

.35805 


.35871 
.35904 
.35937 


.36101 
.36134 
.36167 


.36265 


Cotang 


Cotang 


2.90421 
2.90147 
2.89873 
2.89600 
2.89327 
2.89055 
2.88783 
2.88511 
2.88240 
2.87970 
2.87700 

2.87430 
2.87161 


2.86356 
2.86089 
2.85822 
2.85555 


2.84758 
2.84494 


2.83965 


2.83176 
2.82914 
2.82653 
2.82391 

2.82130 
2.81870 
2.81610 
2.81350 
2.81091 
2.80833 
2.80574 
2.80316 
2.80059 
2.79802 

2.79545 
2.79289 
2.79033 
2.78778 
2.78523 
fc. 78269 
2.78014 
2.77761 
2.77507 
2.77254 

2.77002 
2.76750 
2.76498 
2.76247 
2.75996 
2.75746 
2.75496 
2.75246 
2.74997 
2.74748 


Tang 


71' 


70° 


126  NATURAL   TANGENTS   AND    COTANGENTS. 


20° 

21° 

22°           |            23° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang  1  1  Tang 

Cotang 

/ 

( 

.36397 

2.74748 

.38386 

2.60509 

.40403 

2.47509 

.42447 

2.35585    60 

.36430 

2.74499 

.38420 

2.60283 

.40436 

2.47302 

.42482 

2.35395  i59 

i 

.36463 

2.74251 

.38453 

2.60057 

.40470 

2.47095 

.42516 

2.35205    58 

.36496 

2.74004 

.38487 

2.59831 

.40504 

2.46888 

.42551 

2.35015    57 

< 

.36529 

2.73756 

.38520 

2.59606 

.40538 

2.46682 

.42585 

2.34825    5fi 

J 

.36562 

2.73509 

.38553 

2.59381 

.40572 

2.46476 

.42619 

2.34636 

55 

( 

.36595 

2.73263 

.38587 

2.59156 

.40606 

2.46270 

.42654 

2.34447 

54 

» 

.36628 

2.73017 

.38620 

2.58932 

.40640 

2.46065 

.42688 

2.34258 

53 

8 

.36661 

2.72771 

.38654 

2.58708 

.40674 

2.45860 

.42722 

2.34069 

52 

9 

.36694 

2.72526 

.38687 

2.58484 

.40707 

2.45655 

.42757 

2  33881    51 

10 

.36727 

2.72281 

.38721 

2.58261 

.40741 

2.45451 

.42791 

2.33693  j  50 

11 

.36760 

2.72036 

.38754 

2.58038 

.40775 

2.45246 

.42826 

2  33505    49 

12 

.36793 

2.71792 

.38787 

2.57815 

.40809 

2.45043 

.42860 

2.33317    48 

13 

.36826 

2.71548 

.38821 

2.57593 

.40843 

2.44839 

.42894 

2.33130 

47 

14 

.36859 

2.71305 

38854 

2.57371 

.40877 

2.44636 

.42929 

2.32943    46 

15 

.30892 

2.71062 

.38888 

2.57150 

.40911 

2.44433 

.42963 

2.32756    45 

16 

.36925 

2.70819 

.38921 

2.56928 

.40945 

2.44230 

.42998 

2.32570    44 

17 

.30958 

2.70577 

.38955 

2.56707 

.40979 

2.44027 

.43032 

2.32383    43 

10 

.30901 

2.70335 

.88988 

2.56487 

.41013 

2.43825 

.43067 

2.32197    42 

19 

.37024 

2.70094 

.39022 

2.56266 

.41047 

2.43623 

.43101 

2.32012    41 

2C 

.37057 

2.69853 

.39055 

2.56046 

.41081 

2.43422 

.43136 

2.31826 

40 

21 

.37090 

2.69612 

.39089 

2.55827 

.41115 

2.43220 

.43170 

2.31641 

39 

22 

.37123 

2.C3371 

.39122 

2.55608 

.41149 

2.43019 

.43205 

2.31456  i38 

on 

.37157 

2.0131 

.39156 

2.55389 

.41183 

2.42819 

.43239 

2.31271    37  i 

2J 

.37103 

2.63892 

.39190 

2.55170 

.41217 

2.42618 

.43274 

2.31086    36  : 

^J 

.37223 

2.68G53 

.39223 

2.54952 

.41251 

2.42418 

.43308 

2.30902    35; 

3fl 

.37253 

2.63414 

.39257 

2.54734 

.41285 

2.42218 

.43343 

2.30718    34 

27 

.37289 

2.C3175 

.39290 

2.54516 

.41319 

2.42019 

.43378 

2.30534    33 

23 

.37322 

2.67C37 

.39324 

2.54299 

.41353 

2.41819 

.43412 

2.30351  |32 

C3 

.37355 

2.67700 

.39357 

2.54082 

.41387 

2.41620 

.43447 

2.30167    31 

30 

.37388 

2.67462 

.39391 

2.53865 

.41421 

2.41421 

.43481 

2.29984 

30 

31 

.37432 

2.67225 

.39425 

2.53648 

.41455 

2.41223 

.43516 

2.29801 

29 

32 

.37455 

2.60989 

.39458 

2.53432 

.41490 

2.41025 

.43550 

2.29619  |28 

33 

.37488 

2.66752 

.39492 

2.53217 

.41524 

2.40827 

.43585 

2.29437    27 

34 

.37521 

2.66516 

.39526 

2.53001 

.41558 

2.40629 

.43620 

2.29254  '26 

35 

.37554 

2.66281 

.39559 

2.52786 

.41592 

2.40432 

.43654 

2.29073    25 

36 

.37588 

2.66046 

.39593 

2.52571 

.41626 

2.40235 

.43689 

2.28891    24 

37 

.37621 

2.65811 

.39626 

2.52357 

.41660 

2.40038 

.43724 

2.28710    23  i 

33 

.37654 

2.65576 

.39660 

2.52142 

.41694 

2.39841 

.43758 

2.28528  :22i 

30 

.37687 

2.65342 

.39694 

2.51929 

.41728 

2.39645 

.43793 

2.28348    21  ! 

40 

.37720 

2.65109 

.39727 

2.51715 

.41763 

2.39449 

.43828 

2.28167 

20 

41 

.37754 

2.64875 

.39761 

2.51502 

.41797 

2.89253 

.43862 

2.27987 

191 

42 

.37787 

2.64G42 

.39795 

2.51289 

.41831 

2.39058 

.43897 

2  27806 

18  ! 

43 

.37820 

2.64410 

.39829 

2.51076 

.41865 

2.38863 

.43932 

2.27626 

17 

44 

.37853 

2.64177 

.39862 

2.50864 

41899 

2.38668 

.43966 

2.27447 

16 

45 

.37887 

2.63945 

.39896 

2.50652 

.41933 

2.38473 

.44001 

2.27267 

15 

46 

.37920 

2.63714 

.39930 

2.50440 

.41968 

2.38279 

.44036 

2.27088 

14 

47 

.37953 

2.63483 

.39963 

2.50229 

.42002 

2.38084 

.44071 

2.26909 

18 

48 

.37986 

2.63252 

.39997 

2.50018 

.42036 

2.37891 

.44105 

2.26730 

12 

40 

.33020 

2.63021 

.40031 

2.49807 

.42070 

2.37697 

.44140 

2.26552 

11 

50 

.38053 

2.62791 

.40065 

2.49597 

.42105 

2.37504 

.44175 

2.26374 

10 

51 

.38086 

2.62561 

.40098 

2.49386 

.42139 

2.87311 

.44210 

2.26196 

9 

52 

.33120 

2.62332 

.40132 

2.49177 

.42173 

2.37118 

.44244 

2.26018 

8 

53 

.38153 

2.62103 

.40166 

2.48967 

.42207 

2.36925 

.44279 

2.25840 

7 

54 

.38186 

2.61874 

.40200 

2.48758 

.42242 

2.36733 

.44314 

2.25663 

6 

55 

.38220 

2.61646 

.40234 

2.48549 

.42276 

2.36541 

.44349 

2.25486 

5 

56 

.38253 

2.61418 

.40267 

248340 

.42310 

2.36349 

.44384 

2.25309 

4 

57 

.38286 

2.61190 

.40301 

2.48132 

.42345 

2.36158 

.44418 

2.25132 

3 

58 

.38320 

2.60963 

.40335 

2.47924 

.42379 

2.35967 

.44453 

2.24956 

2 

59 

.38353 

2.60736 

.40369 

2.47716 

.42413 

2.35776 

.44488 

2  24780 

1 

60 

.38386 

2.60509 

.40403 

2.47509 

.42447 

2.85585 

.44523 

2.24604 

0 

/ 

Cotans 

Tang  j 

Cotang     Tang    i;  Cotang 

Tang 

Cotang     Tang 

/ 

i            69° 

)  68°            f 

67°                     66° 

NATURAL  TANGENTS   AND   COTANGENTS. 


| 

4° 

2 

5° 

2 

6» 

2 

7« 

,_* 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

9 

0 

.44523 

2.24604 

.46631 

2.14-151 

.48773 

2.05030 

.50953 

.96261 

60 

1 

.44558 

2.24428 

.40666 

2.14288 

.48809 

2.04879 

.50989 

.96120 

59 

9 

.44593 

2.24252 

.46702 

2.14125 

.48845 

2.04728 

.51026 

.95979 

58 

3 

.44627 

2.24077 

.46737 

2.13963 

.48881 

2.04577 

.51063 

.95838 

57 

4 

.44662 

2.23902 

.46772 

2.12801 

.48917 

2.04426 

.51099 

.95698 

56 

5 

.44697 

2.23727 

.46808 

2.13639 

.48953 

2.04276 

.51136 

.95557 

55 

6 

.44732 

2.23553 

.46843 

2.13477 

.48989 

2.04125 

.51173 

.95417 

54 

7 

.44767 

2.23378 

.46879 

2.13316 

.49026 

2.03975 

.51209 

.95277 

53 

8 

.44802 

2.23204 

.46914 

2.13154 

.49062 

2.03825 

.51246 

.95137 

52 

9 

.44837 

2.23030 

.40950 

2.12993 

.49098 

2.03675 

.51283 

.94997 

51 

10 

^44872 

2.22857 

.46985 

2.12832 

.49134 

2.03526 

.51319 

.94858 

50 

11 

T44907 

2.22683 

.47021 

2.12671 

.49170 

2.03376 

.51356 

.94718 

49 

13 

.44942 

2.22510 

.47056 

2.12511 

.49206 

2.03227 

.51393 

.94579 

48 

13 

.44977 

2.22337 

.47092 

2.12350 

.49242 

2.03078 

.51430 

.94440 

47 

14 

.45012 

2.22164 

.47128 

2.12190 

.49278 

2.02929 

.51467 

.94301 

46 

15 

.45047 

2.21992 

.47163 

2.12030 

.49315 

2.02780 

.51503 

.94162 

45 

10 

.45082 

2.21819 

.47199 

2.11871 

.49351 

2.02631 

.51540 

.94023 

44 

ir 

.45117 

2.21647 

.47234 

2.11711 

.49387 

2.02483 

.51577 

.93885 

43 

18 

.45152 

2.21475 

.47,370 

2.11552 

.49423 

2.02335 

.51614 

.93746 

42 

1!) 

.45187 

2.21304 

.4rC05 

2.11C92 

.49459 

2.02187 

.51651 

.93608 

41 

90 

.45222 

2.21132 

.47341 

2.11233 

.49495 

2.02039 

.51688 

.93470 

40 

21 

.45257 

2.20961 

.47377 

2.11075 

.49532 

2.01891 

.51724 

.93333 

39 

22 

.45292 

2.20790 

.47412 

2.1CD16 

.49568 

2.01743 

.51761 

.93195 

38 

•_"! 

.45327 

2.20619 

.47448 

2.10758 

.49604 

2.01596 

.51798 

.93057 

37 

84 

.45362 

2.20449 

.47483 

8.10600 

.49640 

2.01449 

.51835 

.92920 

38 

25 

.45397 

2.20278 

.47519 

2.10442 

.49677 

2.01302 

.51873 

.92783 

35 

£6 

.45432 

2.20108 

.47555 

2.10234 

.49713 

2.01155 

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.92645 

34 

8? 

.45467 

2.19938 

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2.10126 

.49749 

2.01008 

.51946 

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oo 

28 

.45502 

2.19769 

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2.CC3G9 

.49786 

2.00862 

.51983 

.92371 

32 

89 

.45538 

2.19599 

.47662 

2.09011 

.49822 

2.00715 

.52020 

.92235 

31 

30 

.45573 

2.19430 

.47698 

2.09654 

.49858 

2.00569 

.52057 

.92098 

30 

81 

.45608 

2.19261 

.47733 

2.09408 

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2.00423 

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.91963 

29 

83 

.45643 

2.19092 

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2.09341 

.49931 

2.00277 

.52131 

.91826 

23 

83 

.45678 

2.10923 

.47805 

2.09184 

.49967 

2.00131 

.52168 

:  .91690 

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84 

.45713 

2.18755 

.47840 

2.09028 

.50004 

:  .99986 

.52205 

.91554 

20 

85 

.45748 

2.18587 

.47876 

2.08872 

.50040 

.99841 

.52242 

.91418 

25 

86 

.45784 

2.18419 

.47912 

2.08716 

.50076 

.99695 

.52279 

.91282 

24 

37 

.45819 

2.18251 

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2.08560 

.50113 

.99550 

.52316 

.91147 

23 

38 

.45854 

2.18084 

.47984 

2.08405 

.50149 

.99406 

.52353 

.91012 

22 

89 

.45889 

2.17916 

.019 

2.03250 

.50185 

.99261 

.52390 

.90876 

21 

40 

.45924 

2.17749 

.43055 

2.08094 

.50223 

.99116 

.52427 

.90741 

20 

-11 

.45960 

2.17582 

.48091 

2.07939 

.50258 

.98973 

.52464 

.90607 

19 

43 

.45995 

2.17416 

.43127 

2.07785 

.50295 

.98828 

.52501 

:  .90473 

18 

43 

.4G030 

2.17249 

.48163 

2.07630 

.50331 

.98684 

.52538 

:  .90337 

17 

44 

.40065 

2.17083 

.48198 

2.07476 

.50368 

.98540 

.52575 

:  .90203 

16 

45 

.40101 

2.16917 

.48234 

2.07321 

.50404 

.98396 

.52613 

.90069 

15 

40 

.40136 

2.16751 

.48270 

2.07167 

.50441 

.98253 

.52650 

.89935 

14 

47 

.46171 

2.16585 

.48306 

2.07014 

.50477 

.98110 

.52687 

.89801 

13 

48 

.46206 

2.16420 

.48343 

2.06860 

.50514 

:  .97966 

.52724 

.89667 

12 

40 

.46242 

2.10255 

.48378 

2.06706 

.50550 

.97823 

.52761 

.89533 

11 

50 

.46277 

2.16090 

.48414 

2.06553 

.50587 

.97681 

.52798 

.89400 

10 

51 

.46312 

2.15925 

.48450 

2.06400 

.50623 

.97538 

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.89266 

9 

59 

.46348 

2.15760 

.48486 

2.06247 

.50660 

.97395 

.52873 

.89133 

8 

53 

.46383 

2.15596 

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2.06094 

.50696 

.97253 

.52910 

.89000 

7 

54 

.46418 

2.15432 

.48557 

2.05942 

.50733 

.97111 

.52947 

.88867 

6 

53 

.46454 

2.15268 

.48593 

2.05790 

.50769 

.96969 

.52985 

.88734 

5 

no 

.46489 

2.15104 

.48629 

2.05637 

.50806 

.96827 

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:  .88602 

4 

57 

.46525 

2.14940 

.48665 

2.05485 

.50843 

.96685 

.53059 

:  .88469 

3 

58 

.46560 

2.14777 

.48701 

2.05333 

.50879 

:  .96544 

.53096 

.88337 

2 

59 

.46595 

2.14614 

.48737 

2.05182 

.50916 

1.96402 

.53134 

.88205 

1 

60 

.46631 

2.14451 

.48773 

2.05030 

.50953 

1.96261 

.53171 

.88073 

C 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

f 

6 

5° 

6 

4° 

6 

3° 

6 

2*         .1 

128         NATURAL  TANGENTS   AND   COTANGENTS, 


f 

28° 

29°            i           30« 

31° 

Tang 

Cotang 

Tang 

Cotang 

Tang     Cotang 

Tang 

Cotang 

"6 

.53171 

1.88073 

.55431 

1.80405 

IT57735" 

1.73205 

.60086 

1.60428 

60; 

i 

.53208 

1.87941 

.55469 

1.80281 

1   .57774 

1.73089 

.60126 

1.66318 

59  > 

o 

.53246 

1.87809 

.55507 

1.80158 

.57813 

1.72973 

.60165 

1.66209 

58 

3 

.53283 

1.87677 

.55545 

1.80034 

.57851 

1.72857 

.60205 

1.66099 

57 

4 

.53320 

1.87546 

.55583 

1.79911 

;   .57890 

1.72741 

.60245 

1.65990 

50 

g 

.53358 

1.87415 

.55621 

1.79788 

1   .57929 

1.72625 

.60284 

1.65881 

55 

e 

.53395 

1.87283 

.55659 

1.79665 

.57968 

1.72509 

.60324 

1.65772 

54 

7 

.53432 

1.87152 

.55697 

1.79542 

.58007 

1.72393 

.60364 

1.65663 

53 

8 

.53470 

1.87021 

.55736 

1.79419  ;i   .58046 

1.72278 

.60403 

1.65554 

52 

9 

.53507 

1.86891 

.55774 

1.79296 

.58085 

1.72163 

.60443 

1.65445 

51 

10 

.53545 

1.86760 

.55813 

1.79174 

.58124 

1.72047 

.60483 

1.65337 

50 

11 

.53582 

1.86630 

.55850 

1.79051 

.58162 

1.71932 

.60522 

1.65228 

49 

12 

.53620 

1.86499 

.55888 

1.78929 

.58201 

1.71817 

.60562 

1.65120 

48 

13 

.53657 

1.86369 

.55926 

1.78807 

.58240 

1.71702 

.60602 

1.65011 

47 

14 

.53694 

1.86239 

.55964 

1,78685 

.58279 

1.71588 

.60642 

1.64903 

46 

15 

.53732 

1.86109 

.56003 

1.78563 

.58318 

1.71473 

.60681 

1.64795 

45 

16 

.53769 

1.85979 

.56041 

1.78441 

.58357 

1.71358 

.60721 

1.64687 

44 

17 

.53807 

1.85850 

.56079 

1.78319 

.58396 

1.71244 

.60761 

1.64579 

43 

18 

.53844 

1.85720 

.56117 

1.78198 

.58435 

1.71129 

.60801 

1.64471 

42 

19 

.53882 

1.85591 

.56156 

1.78077 

.58474 

1.71015 

.60841 

1.64363 

41 

20 

.53920 

1.85462 

.56194 

1.77955 

.58513 

1.70901 

.60881 

1.64256 

40 

21 

.53957 

1.85333 

.56232 

1.77834 

.58552 

1.70787 

.60921 

1.64148 

39 

•J-.' 

.53995 

1.85204 

.56270 

1.77713 

.58591 

1.70073 

.60960 

1.64041 

38 

-,>:; 

.54032 

1.85075 

.56309 

-1.77592 

i   .58631 

1.70560 

.61000 

1.63934 

37 

24 

.54070 

1.84946 

.56347 

1.77471 

.58670 

1.70446 

.61040 

1.63826 

36 

25 

.54107 

1.84818 

.56385 

1.77351 

.58709 

1.70332 

.61080 

1.63719 

35 

36 

.54145 

1.84689 

.56424 

1.77230 

'   .58748 

1.70219 

.61120 

1.63612 

34 

27 

.54183 

1.84561 

.56463 

1.77110 

.58787 

1.70106 

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1.63505 

33 

28 

.54220 

1.84433 

.56501 

1.76990 

.58826 

1.69992 

.61200 

1.63398 

32 

',",) 

.54258 

1.84305 

.5C539 

1.7G869 

.58865 

1.69879 

.61240 

1.63292 

31 

30 

.54296 

1.84177 

.56577 

1.76749 

.58905 

1.69766 

.61280 

1.63185 

30 

31 

.54333 

1.84049 

.56616 

1.76629 

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1.69653 

.61320 

1.63079 

29 

32 

.54371 

1.83922 

.50654 

1.70510 

.58983 

1.69541 

.61360 

1.62972 

28 

83 

.54409 

1.83794 

.56693 

1.76390 

.59022 

1.69428 

.61400 

1.62866 

27 

Ml 

.54446 

1.83667 

.56731 

1.70271 

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1.69316 

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1.62760 

26 

-•55 

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1.83540 

.56769 

1.70151 

.59101 

1.69203 

.61480 

1.62654 

25 

36 

.54528 

1.83413 

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1.76032 

:   .59140 

1.69091 

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1.62548 

24 

37 

.54560 

1.83286 

.56846 

1.75913 

!   .59179 

1.68979 

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1.62442 

23 

38 

.54597 

1.  83159 

.56885 

1.75794 

]   .59218 

1.68866 

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1.C2336 

22 

39 

.54635 

1.83033 

.56923 

1.75675 

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1.68754 

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1.62230 

21 

-10 

.54673 

1.62906 

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1.75556 

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1.68643 

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1.02125 

20 

41 

.54711 

1.82780 

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1.75437 

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1.68531 

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1.62019 

19 

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.54748 

1.82654 

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1.75319 

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1.68419 

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1.61914 

18 

43 

.54786 

1.82528 

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1.75200 

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1.68308 

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1.61808 

17 

44 

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1.82402 

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1.75082 

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1.68196 

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1.61703 

16 

15 

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1.82276 

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1.74964 

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1.68085 

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1.61598 

15 

16 

.54900 

1.82150 

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1.74846 

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1.67974 

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1.61493 

14 

47 

.54938 

1.82025 

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1.74728 

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1.67863 

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1.61388 

13 

48 

.54975 

1.81899 

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1.74610 

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1.67752 

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1.61283 

12 

•19 

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1.81774 

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1.74492 

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1.67641 

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1.61179 

11 

50 

.55051 

1.81649 

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1.74375 

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1.67530 

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1.61074 

10 

51 

.55089 

1.81524 

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1.74257 

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1.67419 

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1.60970 

9 

52 

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1.81399 

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1.74140 

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1.67309 

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1.60865 

8 

53 

.55165 

1.81274 

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1.74022 

.59809 

1.67198 

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1.60761 

7 

54 

.55203 

1.81150 

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1.73905 

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1.67088 

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1.60657 

6 

55 

.55241 

1.81025 

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1.73788 

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1.66978 

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1.60553 

5 

56 

.55279 

1.80901 

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1.73671 

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1.66867 

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1.60449 

4 

57 

.55317 

1.80777 

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1.73555 

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1.66757 

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1.60345 

3 

58 

.55355 

1.80653 

.57657 

1.73438 

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1.66647 

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1.60241 

2 

59 

.55393 

1.80529 

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1.73321 

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1.66538 

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1.60137 

1 

no 

.554'il 

1.80405 

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1.73505 

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I.fifi428 

.02487 

1.60038 

0 

/ 

Cotang 

Tang      Cotang     Tang 

Cotang 

Tang 

Cotaug      Tang 

9 

61°                      60°                      59' 

58° 

NATURAL  TANGENTS   AND   COTANGENTS. 


129 


32 

o 

33 

0 

34 

0 

35 

o 

Tang 

otang 

Tang 

otang 

Tang 

Cotang 

Tang 

Cotang 

8    .62487 

.000*5 

.64941 

.53986 

67451 

.48256 

70021 

.42815 

jO 

62527 

.59930 

.64982 

.53888 

67493 

.48163 

70064 

1.42720 

9 

2     .62568 

.59826 

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.53791 

67536 

.48070  1 

70107 

.42638 

8 

3    .62608 

.59723 

.65065 

.53693 

67578 

.47977 

70151 

1.42550 

7 

4    .62649 

.59620 

65106 

.53595 

67620 

.47885 

70194 

1.42462 

5 

5    .62689 

.59517 

65148 

.53497 

67663 

.47792 

70238 

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3 

6      62730 

.59414 

65189 

.53400 

67705 

1.47699  ' 

70281 

[.42286 

4 

7    .62770 

.59311 

.65231 

.53302 

67748 

.47607  j 

70325 

1.42198 

3 

8    .62811 

.59208 

.65272 

.53205 

67790 

1.47514 

.70368 

1.42110 

2 

9    .62852 

.59105 

.65314 

.53107 

67832 

..47422 

70412 

1.42022 

1 

10    .62893 

.59002 

.65355 

.53010 

67875 

1.47330 

70455 

1.41934 

50 

11    .62933 

.58900 

.65397 

.52913 

67917 

1.47238 

.70499 

1.41847 

9 

12    .62973 

.53797 

.65438 

.52816 

67960 

..47146 

.70542 

1.41759 

8 

13    .63014 

.58695 

.65480 

.52719 

68002 

[.47053 

.70586 

1.41672 

7 

14     .63055 

.58593 

.65521 

.52622 

68045 

1.46962 

.70629 

1.41584 

6 

15    .63095 

.58490 

.65563 

.52525 

68088 

1.46870 

.70673 

1.41497 

5 

16    .63136 

.58388 

.65604 

.52429 

.68130 

[.46778 

.70717 

1.41409 

44 

17    .63177 

.58286 

.65646 

.52332 

.68173 

[.46686 

.70760 

1.41322 

3 

18    .63217 

.58184 

.65688 

1.52235 

.68215 

1.46595 

.70804 

1.41235 

42 

19      63258 

.58083 

.65729 

1.52139 

.68258 

1.46503 

.70848 

1.41148 

1 

20    .63299 

1.57981 

.65771 

1.53043 

.68301 

1.46411 

.70891 

1.41061 

40 

21    .63340 

1.57879 

.65813 

1.51946 

.68343 

1.46320 

.70935 

1.40974 

39 

22     .63380 

1.57773 

.65854 

:.  51850 

.68386 

1.46229 

.70979 

1.40887 

38 

23     .63421 

1.57G76 

.65896 

1.51754 

.68429 

1.46137 

.71023 

1.40800 

17 

24     .63462 

1.57575 

.65938 

[.51658 

.68471 

1.46046 

.71066 

1.40714 

IG 

25     .63503 

1.57474 

.65980 

[.515G2 

.68514 

1.45955 

.71110 

1.40627 

IE 

26     .63544 

1.57372 

.66021 

1.51466 

.68557 

1.45864 

.71154 

1.40540 

34 

27    .63584 

1.57271 

.66063 

1.51370 

.68600 

1.45773 

.71198 

1.40454 

J£ 

28    .63625 

1.57170 

.66105 

1.51275 

.68642 

1.45682 

.71242 

1.40367 

32 

29    .63666 

1.57069 

.66147 

1.51179 

.68685 

1.45592 

.71285 

1.40281 

31 

30    .63707 

1.56969 

.66189 

1.51084 

.68728 

1.45501 

.71329 

1.40195 

30 

31     .63748 

1.56868 

.66230 

1.50988 

.68771 

1.45410 

.71373 

1.40109 

Of 

33    .63789 

1.56767 

.66272 

1.50893 

.68814 

1.45320 

.71417 

1.40022 

28 

33    .63830 

1.56667 

.66314 

1.50797 

.68857 

1.45229 

.71461 

1.39936 

2' 

34    .63871 

1.56566 

.66356 

1.50702 

.68900 

1.45139 

.71505 

1.39850 

26 

35    .63912 

1.56466 

.66398 

1.50607 

.68942 

1.45049 

.71549 

1.39764 

25 

33    .63953 

1.56366 

.66440 

1.50512 

.68985 

1.44958 

.71593 

1.39679 

24 

37    .63994 

1.56265 

.66482 

1.50417 

.69028 

1.44868 

.71637 

1.39593 

2< 

38    .64035 

1.56165 

.66524 

1.50322 

.69071 

1.44778 

.71681 

1.39507 

3$ 

39    .64076 

1.56065 

.66566 

1.50228 

.69114 

1.44688 

.71725 

1.39421 

21 

40    .64117 

1.55966 

.66608 

1.50133 

.69157 

1.44598 

.71769 

1.39336 

20 

41    .64158 

1.55866 

.66650 

1.50038 

.69200 

1.44508 

.71813 

1.39250 

10 

42    .64199 

1.55766 

.66692 

1.49944 

.6924S 

1.44418 

.71857 

1.89165 

1C 

43     .64240 

1.575666 

.66734 

1.49849 

.69286 

1.44329 

.71901 

1.39079 

44    .64281 

1.55567 

.66776 

1.49755 

.69329 

1.4*239 

.71946 

1.38994 

45    .64322 

1.55467 

.66818 

1.49661 

.69372 

1.44149 

.71990 

1.38909 

46    .64363 

1.55368 

.66860 

1.49566 

.694N 

1.44060 

.72034 

1.38824 

47    .64404 

1.55269 

.66902 

1.49472 

.69459 

1.43970 

.72078 

1.38738 

48    .64446 

1.55170 

.66944 

1.49378 

.69502 

1.43881 

.72122 

1.38G53 

49    .64487 

1.55071 

.66986 

1.49284 

.69545 

1.43792 

.72167 

1.38568 

50    .64528 

1.54972 

.67028 

1.49190 

.69588 

1.43703 

.72211 

1.38484 

51    .64569 

1.54873 

.67071 

1.49097 

.69631 

1.43614 

.72255 

1.38399 

52    .6461 

1.54774 

.67113 

1.49003 

.69675 

1.43525 

.72299 

1.38314 

53    .6465 

1  .54675 

.67155 

1.48909 

.69718 

1.43436 

.72344 

1.38229 

54    .6469 

1.5457 

.67197 

1.48816 

.69761 

1.43347 

.72388 

1.38145 

55*    .64734 

1.5447 

.67239 

1.48722 

.69804 

1.43258 

.72432 

1.380GO 

56    .6477 

1.5437 

.67282 

1.48629 

.69847 

1.43169 

.72477 

1.37976 

57    .6481 

1.5428: 

.6732^ 

1.48536 

.69891 

1.43080 

.7252: 

1.37891 

58    .6485 

1.54183 

.6736 

1.48442 

.69934 

1.42992 

.72565 

1.37807 

£9    .6489 

1.5408 

.6740 

1.4834J 

.6997 

1.42905 

.72610 

1.37722 

60    .6494 

1.53986 

.6745 

1.48256 

.7002 

1.42815 

.72654 

1.37638 

Cotan 

Tang 

Cotan 

Tang 

Cotan 

Tang 

j  Cotan 

Tang 

57° 

56° 

53° 

1 

540     2 

130          NATUtlAL   TANGENTS   ANT)    COTANGENTS. 


; 

3 

B° 

3 

7°           I 

3 

5° 

$ 

9° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.72654 

1.37638 

.75355 

1.32704 

.78129 

1.27994 

.80978 

1.23490 

GO 

1 

.72699 

1.37554 

.75401 

1.32624 

.78175 

1.27917 

.81027 

1.23416 

59 

2 

.72743 

1.37470 

.75447 

1.32544 

.78222 

1.27841 

.81075 

1.23343 

58 

3 

.72788 

1.37386 

.75492 

1.32464 

.78269 

1.27764 

.81123 

1.23270 

57 

4 

.72832 

1.37302 

.75538 

1.32384 

.78316 

1.27688 

.81171 

1.23196 

5C 

5 

.72877 

1.37218 

.75584 

1.32304 

.78363 

1.27611 

.81220 

1.23123 

55 

6 

.72921 

1.37134 

.75629 

1.32224 

.78410 

1.27535 

.81268 

1.23050 

54 

7 

.72966 

1.37050 

.75675 

1.32144 

.78457 

1.27458 

.81316 

1.22977 

53 

8 

.73010 

1.36967 

.75721 

1.32064 

.78504 

1.27382 

.81364 

1.22904 

52 

9 

.73055 

1.36883 

.75767 

1.31984 

.78551 

1.27306 

.81413 

1.22*31 

51 

10 

.73100 

1.36800 

.75812 

1.31904 

.78598 

1.27230 

.81461 

1.22758 

50 

11 

.73144 

1.36716 

.75858 

1.31825 

.78645 

1.27153 

.81510 

1.22685 

40 

13 

.73189 

1.36633 

.75904 

1.31745 

.78692 

1.27077 

.81558 

1.22612 

48 

13 

.73234 

1.36549 

.75950 

1.31666 

.78739 

1.27001 

.81606 

1.22539 

47 

14 

.73278 

1.36466 

.75996 

1.31586 

.78786 

1.26925 

.81655 

1.22467 

4G 

15 

.73323 

1.36383 

.76042 

1.31507 

.78834 

1.26849 

.81703 

1.22394 

45 

16 

.73368 

1.36300 

.76088 

1.31427 

.78881 

1.26774 

.81752 

1.22321 

44 

1? 

.73413 

1.36217 

.76134 

1.31348 

.78928 

1.26698 

.81800 

1.22249 

43 

18 

.73457 

1.36134 

.76180 

1.31269 

.78975 

1.26622 

.81849 

1.22176 

42 

19 

.73502 

1.36051 

.76226 

1.31190 

.79022 

1.26546 

.81898 

1.22104 

41 

SO 

.73547 

1.35968 

.76273 

1.31110 

.79070 

1.26471 

.81946 

1.22031 

40 

81 

.73592 

1.35885 

.76318 

1.31031 

.79117 

1.26395 

.81995 

1.21959 

39 

J2 

.73637 

1.35802 

.76364 

1.30952 

.79164 

1.26319 

.82044 

1.21886 

38 

.'3 

.73681 

1.35719 

.76410 

1.30873 

.79212 

1.26244 

.82092 

1.21814 

37 

24 

.73726 

1.35637 

.76456 

1.30795 

.79259 

1.26169 

.82141 

1.21742 

3C 

25 

.73771 

1.35554 

.76502 

1.30716 

.79306 

1.26093 

.82190 

1.21670 

35 

86 

.73816 

1.35472 

.76548 

1.30637 

.79354 

1.26018 

.82238 

1.21598 

34 

87 

.73861 

1.35389 

.76594 

1.30558 

.79401 

1.25943 

.82287 

1.21526 

33 

88 

.73906 

1.35307 

.76640 

1.30480 

.79449 

1.25867 

.82336 

1.21454 

go 

X!!) 

.73951 

1.35224 

.76686 

1.30401 

.79496 

1.25792 

.82385 

1.21382 

81 

30 

.73996 

1.35142 

.76733 

1.30323 

.79544 

1.25717 

.82434 

1.21310 

30 

31 

.74041 

1.35060 

.76779 

1.30244 

.79591 

1.25642 

.82483 

1.21238 

20 

32 

.74086 

1.34978 

.76825 

1.30166 

.79639 

1.25567 

.82531 

1.21166 

28 

88 

.74131 

1.34896 

.76871 

1.30087 

.79680 

1.25492 

.82580 

1.21094 

27 

34 

.7417S 

1.34814 

.76918 

1.30009 

.79734 

1.25417 

.82629 

1.21023 

26 

35 

.74221 

1.34732 

.76964 

1.29931 

.79781 

1.25343 

.82678 

1.20951 

25 

36 

.74267 

1.34650 

.77010 

1.29853 

.79829 

1.25268 

.82727 

1.20879 

24 

87 

.74312 

1.34568 

.77057 

1.29775 

.79877 

1.25193 

.82776 

1.20808 

23 

38 

.74357 

1.34487 

.77103 

1.29696 

.79924 

1.25118 

.82825 

1.20736 

22 

39 

.74402 

1.34405 

.77149 

1.29618 

.79972 

1.25044 

.82874 

1.20665 

21 

40 

.74447 

1.343S3 

.77196 

1.29541 

.80020 

1.24969 

.82923 

1.20593 

20 

41 

.74492 

1.34242 

.77243 

1.29463 

.80067 

1.24895 

.82972 

1.20522 

19 

43 

.74538 

1.34160 

.77289 

1.29385 

.80115 

1.24820 

.83022 

1.20451 

IB 

13 

.74583 

1.34079 

.77385 

1.29307 

.80163 

1.24746 

.83071 

1.20379 

17 

44 

.74628 

1.33998 

.77382 

1.29229 

.80211 

1.24672 

.831CO 

1.20308 

16 

45 

.74674 

1.33916 

.77428 

1.29152 

.80258 

1.24597 

.83169 

1.20237 

15 

46 

.74719 

1.33835 

.77475 

1.29074 

.80306 

1.24523 

.83218 

1.20166 

14 

47 

.74764 

1.33754 

.77521 

1.28997 

.80354 

1.24449 

.83268 

1.20095 

13 

48 

.74810 

1.33673 

.77568 

1.28919 

.80402 

1.24375 

.83317 

1.20024 

12 

49 

.74855 

1.33592 

.77615 

1.28842 

1  .80450 

1.24301 

.83366 

1.19953 

11 

no 

.74900 

1.33511 

.77661 

1.28764 

.80498 

1.24227 

.83415 

1.19882 

10 

51 

.74946 

1.33430 

.77708 

1.28687 

.80546 

1.24153 

.83465 

1.19811 

9 

52 

.74991 

1.33349 

.77754 

1.28610 

.80594 

1.24079 

.83514 

1.19740 

8 

53 

.75037 

1.33268 

.77801 

1.28533 

.80642 

1.24005 

.85564 

1.19669 

7 

54 

.75082 

1.33187 

.77848 

1.28456 

.80690 

1.23931 

.83613 

1.19599 

6 

55 

.75128 

1.33107 

.77895 

1.28379 

.80738 

1.23858 

.83662 

1.19528 

5 

50 

.75173 

1.33026 

.77941 

1.28302 

.80786 

1.23784 

.83712 

1.19457 

4 

57 

.75219 

1.32946 

.77988 

1.28225 

.80834 

1.23710 

.83761 

1.19387 

3 

58 

.75204 

1.32865 

.78035 

1.28148 

.80882 

1.23637 

.83811 

1.19316 

2 

59 

.75310 

1.32785 

.78082 

1.28071 

,80930 

1.23563 

.83860 

1.19246 

1 

00 

.75355 

1.32704 

.78129 

1.27994 

.80978 

1.23490 

.83910 

1.19175 

0 

t 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

( 

)3° 

i 

(2° 

6 

1« 

| 

0° 

NATURAL  TANGENTS  AND  COTANGENTS. 


131 


40° 

41o 

42" 

43° 

Tang 

Cotang 

Tang     Cotang 

Tang 

Cotang 

Tang 

Cotang 

1 

(' 

.83910 

1.19175 

.86929 

1.15037 

.90040 

1.11061 

.93252 

1.07237 

60 

.83960 

1.19105 

.86980 

1.14969 

.90093 

1  10996 

.93306 

1.07174 

59 

£ 

.84009 

1.19035 

.87031 

1.14902 

.90146 

1.10931 

.93360 

1.07112 

58 

8 

.84059 

1.18964 

.87082 

1.148S* 

.90199 

1.10867 

.93415 

1.07049 

57 

4 

.84108 

1.18894 

.87133 

1.14767 

.90251 

1.10802 

.93469 

1.06987 

56  i 

5 

.84158 

1.18824 

.87184 

1.14699 

.90304 

1.10737 

.93524 

1.06925 

55 

6 

.84208 

1.18754 

.87236 

1.14632 

.90357 

1.10672 

.93578 

1.06862 

54! 

7 

.84258 

1.18684 

.87287 

1.14565 

.90410 

1.10607 

.93633 

1.06800 

53  1 

H 

.84307 

1.18614 

.87338 

1.14498 

.90463 

1.10543 

.93688 

1.06738 

52! 

9 

.84357 

1.18544 

.87389 

1.14430 

.90516 

1.10478 

.93742 

1.06676 

51 

JO 

.84407 

1.18474 

.87441 

1.14363 

.90569 

1.10414 

.93797 

1.06613 

50 

11 

.84457 

1.18404 

.87492 

1.14296 

.00621 

1.10349 

.93852 

1.06551 

49 

19 

.84507 

1.18334 

.87543 

1.14229 

.90674 

1.10285 

.93906 

1.06489 

48 

18 

.84556 

1.18264 

.87595 

1.14162 

.90727 

1.10220 

.93961 

1.06427 

47 

14 

.84606 

1.18194 

.87646 

1.14095 

.90781 

1.10156 

.94016 

1.06365 

46 

15 

.84656 

1.18125 

.87698 

1.14028 

.90834 

1.10091 

.94071 

1.06303 

45 

16 

.84706 

1.18055 

.87749 

1.13961 

.90887 

1.10027 

.94125 

1.06241 

44 

17 

.84756 

1.17986 

.87801 

1.13894 

.90940 

1.09963 

.94180 

1.06179 

43 

18 

.84306 

1.17916 

.87852 

1.13828 

.90993 

1.09899 

.94235 

1.06117 

42 

19 

.84856 

1.17846 

.87904 

1.13761 

.91046 

1  .09&34 

.94290 

1.06056 

41 

80 

.84906 

1.17777 

.87955 

1.13694 

.91099 

1.09770 

.94345 

1.05994 

40 

81 

.84956 

1.17708 

.88007 

1.13627 

.91153 

1.09706 

.94400 

1.05932 

39 

23 

.85006 

1.17638 

.88059 

1.13561 

.91206 

1.09642 

.94455 

1.05870 

38 

88 

.85057 

1.17569 

.88110 

1.13494 

.91259 

1.09578 

.94510 

1.05809 

37 

24 

.85107 

1.17500 

.88162 

1.13428 

.91313 

1.09514 

.94565 

1.05747 

36 

85 

.85157 

1.17430 

.88214 

1.13361 

.91366 

1.09450 

.94620 

1.05685 

35 

26 

.85207 

1.17361 

.88265 

1.13295 

.91419 

1.09386 

.94676 

1.05624 

34 

87 

.85257 

1.17292 

.88317 

1.13228 

.91473 

1.09322 

.94731 

1.05562 

33 

88 

.85308 

1.17223 

.88369 

1.13162 

.91526 

1.09258 

.94786 

1.05501 

32 

89 

.85358 

1.17154 

.88421 

1.13096 

.91580 

1.09195 

.94841 

1.05439 

31 

30 

.85408 

1.17085 

.88473 

1.13029 

.91633 

1.09131 

.94896 

1.05378 

30 

81 

.85458 

1.17016 

.88524 

1.12963 

.91687 

1.09067 

.94952 

1.05317 

29 

:w 

.85509 

1.16947 

.88576 

1.12897 

.91740 

1.C9003 

.95007 

1.05255 

28 

:::: 

.85559 

1.16878 

.88628 

1.12831 

.91794 

1.CG940 

.95062 

1.05194 

27 

84 

.85609 

1.16809 

.88680 

1.12765 

.91847 

1.08876 

.95118 

1.05133 

26 

86 

.85660 

1.16741 

.88732 

1.12699 

.91901 

1.08813 

.95173 

1.05072 

25 

86 

.85710 

1.16672 

.88784 

1.12633 

.91955 

1.C8749 

.95229 

1.05010 

24 

87 

.85761 

1.16603 

.88836 

1.12567 

.92008 

1.08686 

.95284 

1.04949 

23 

88 

.85811 

1.16535 

.88888 

1.12501 

.92062 

1.08622 

.95340 

1.04888 

22 

::<) 

.85862 

1.16466 

.88940 

1.12435 

.92116 

1.03559 

.95395 

1.04827 

21 

40 

.85912 

1.16398 

.88992 

1.13369 

.92170 

1.08496 

.95451 

1.04766 

20 

41 

.85963 

1.16329 

.89045 

1.12303 

.92224 

1.08432 

.95506 

1.04705 

19 

i;3 

.86014 

1.16261 

.89097 

1.12238 

.92277 

1.C33G9 

.95562 

1.04644 

18 

(3 

.86064 

1.16192 

.89149 

1.12172 

.92331 

1.03306 

.95618 

1.04583 

17 

44 

.86115 

1.16124 

.89201 

1.12106 

.92385 

1.08243 

.95673 

1.04522 

16 

45 

.86166 

1.16056 

.89253 

1.12041 

.92439 

1.08179 

.95729 

1.04461 

15 

46 

.86216 

1.15987 

.89306 

1.11975 

.92493 

1.08116 

.95785 

1.04401 

14 

47 

.66267 

1.15919 

.89358 

1.11909 

.92547 

1.08053 

.95841 

1.04340 

13 

48 

.86318 

1.15851 

.89410 

1.11844 

.92601 

1-07990 

.95897 

1.04279 

12 

49 

.863G8 

1.15783 

.89463 

1  11778 

.92655 

1.07927 

.95S52 

1.04218 

11 

60 

.86419 

1.15715 

.89515 

1.11713 

.92709 

1.07864 

.96008 

1.04158 

10 

61 

.86470 

1.15647 

.89567 

1.11648 

.92763 

1.07801 

.96064 

1.04097 

9 

68 

.86521 

1.15579 

.89620 

1.11582 

.92817 

1.07738 

.96120 

1.04036 

8 

68 

.86572 

1.15511 

.89672 

1.11517 

.92872 

1.07676 

.96176 

1.03976 

7 

54 

.86623 

1.15443 

.89725 

1.11452 

.92926 

1.07613 

.96232 

1.03915 

6 

65 

.86674 

1.15375 

.89777 

1.11387 

.92980 

1.07550 

.96288 

1.03855 

5 

66 

.86725 

1.15308 

.89830 

1.11321 

.93034 

1.07487 

.96344 

1.03794 

4 

57 

.86776 

1.15240 

.89883 

1.11256 

.93088 

1.07425 

.96400 

1.03734 

3 

68 

.86827 

1.15172 

.89935 

1.11191 

.93143 

1.07362 

.96457 

1.03674 

2 

69 

.86878 

1.15104 

.89988 

1.11126 

.93197 

1.07299 

.96513 

1.03613 

1 

(K 

.86929 

1.15037 

.90040 

1.11061 

.93252 

1.07237 

.96569 

1.03553 

0 

t 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

» 

49° 

48° 

47° 

46° 

132 


NATURAL  TANGENTS  AND  COTANGENTS. 


i 

4 

40 

4 

4« 

4 

40 

t 

t 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.96569 

1.03553 

60 

20 

.97700 

1.02355 

40 

i40 

.98843 

.01170 

20 

1 

.96625 

1.03493 

59 

21 

.97756 

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SHORT-TITLE     CATALOGUE 

OP  THE 

PUBLICATIONS 

OF 

JOHN   WILEY   &    SONS, 

NEW  YORK. 
LONDON:   CHAPMAN  &  HALL,  LIMITED. 


ARRANGED  UNDER  SUBJECTS 


Descriptive  circulars  sent  on  application.  Books  marked  with  an  asterisk  (*)  are  sold 
at  net  prices  only,  a  double  asterisk  (**)  books  sold  under  the  rules  of  the  American 
Publishers'  Association  at  net  prices  subject  to  an  extra  charge  for  postage.  All  books 
are  bound  in  cloth  unless  otherwise  stated. 


AGRICULTURE. 

Armsby's  Manual  of  Cattle-feeding lanao,  $i  75 

Principles  of  Animal  Nutrition 8vo,  4  oo 

Budd  and  Hansen's  American  Horticultural  Manual: 

Part  I.  Propagation,  Culture,  and  Improvement i2mo,  i  50 

Part  II.  Systematic  Pomology i2mo,  i  50 

Downing's  Fruits  and  Fruit-trees  of  America 8vo,  5  oo 

Elliott's  Engineering  for  Land  Drainage i2mo,  i  50 

Practical  Farm  Drainage i2mo,  i  oo 

Green's  Principles  of  American  Forestry i2mo,  i  50 

Grotenfelt's  Principles  of  Modern  Dairy  Practice.      (Woll.) i2mo,  2  oo 

Kemp's  Landscape  Gardening i2mo,  2  50 

Maynard's  Landscape  Gardening  as  Applied  to  Home  Decoration i2mo,     i   *o 

Sanderson's  Insects  Injurious  to  Staple  Crops i2mo,  i  50 

Insects  Injurious  to  Garden  Crops.     (In  preparation.) 
Insects  Injuring  Fruits.     (In  preparation.) 

Stockbridge's  Rocks  and  Soils 8vo,  2  50 

Woll's  Handbook  for  Farmers  and  Dairymen i6mo,  i  50 

ARCHITECTURE. 

Baldwin's  Steam  Heating  for  Buildings i2mo,  2  50 

Berg's  Buildings  and  Structures  of  American  Railroads 4to,  5  oo 

Birkmire's  Planning  and  Construction  of  American  Theatres 8vo,  3  oo 

Architectural  Iron  and  Steel 8vo,  3  50 

Compound  Riveted  Girders  as  Applied  in  Buildings 8vo,  2  oo 

Planning  and  Construction  of  High  Office  Buildings. 8vo,  3  50 

Skeleton  Construction  in  Buildings 8vo,  3  oo 

Brigg's  Modern  American  School  Buildings 8vo,  4  oo 

Carpenter's  Heating  and  Ventilating  of  Buildings 8vo,  4  oo 

Freitag's  Architectural  Engineering 8vo,  3  50 

Fireproofing  of  Steel  Buildings 8vo,  2  50 

French  and  Ives's  Stereotomy.  , 8vo,  2  50 

Gerhard's  Guide  to  Sanitary  House-inspection i6mo,  i  oo 

Theatre  Fires  and  Panics i2mo,  i  50 

Holly's  Carpenters'  and  Joiners'  Handbook i8mo,  75 

Johnson's  Statics  by  Algebraic  and  Graphic  Methods 8vo,  2  oo 

1 


Kidder's  Architects' and  Builders' Pocket-book.  Rewritten  Edition.  i6mo,mor,,  5  oo 

Merrill's  Stones  for  Building  and  Decoration 8vo,  5  oo 

Non-metallic  Minerals:    Their  Occurrence  and  Uses 8vo,  4  oo 

Monckton's  Stair-building 4to,  4  oo 

Patton's  Practical  Treatise  on  Foundations 8vo,  5  oo 

Peabody's  Naval  Architecture Svo,  7  50 

Richey's  Handbook  for  Superintendents  of  Construction i6mo,  rnor.,  4  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish Svo,  3  oo 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry Svo,  i  SG 

Snow's  Principal  Species  of  Wood Svo,  3  50 

Sondericker's  Graphic  Statics  with  Applications  to  Trusses,  Beams,  and  Arches. 

Svo,  2  oo 

Towne's  Locks  and  Builders'  Hardware iSmo,  morocco,  3  oo 

Wait's  Engineering  and  Architectural  Jurisprudence Svo,  6  oo 

Sheep,  6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  Svo,  5  co 

Sheep,  5  50 

Law  of  Contracts Svo,  3  oo 

Wood's  Rustless  Coatings:   Corrosion  and  Electrolysis  of  Iron  and  Steel.  .Svo,  4  oa 

Woodbury's  Fire  Protection  of  Mills Svo,  2  50 

Worcester  and  Atkinson's  Small  Hospitals,  Establishment  and  Maintenance, 
Suggestions  for  Hospital  Architecture,  with  Plans  for  a  Small  Hospital. 

i2mo,  i   25 

The  World's  Columbian  Exposition  of  1893 Large  4to,  i   oo 

ARMY  AND  NAVY. 

Bernadou's  Smokeless  Powder,  Nitro-cellulose,  and  the  Theory  of  the  Cellulose 

Molecule i2mo,  2  50 

*  Bruff 's  Text-book  Ordnance  and  Gunnery Svo,  6  oo 

Chase's  Screw  Propellers  and  Marine  Propulsion Svo,  3  oo 

Cloke's  Gunner's  Examiner.     (In  press.) 

Craig's  Azimuth 4to,  3  50 

Crehore  and  Squier's  Polarizing  Photo-chronograph Svo,  3  oo 

Cronkhite's  Gunnery  for  Non-commissioned  Officers 24tno,  morocco,  2  co 

*  Davis's  Elements  of  Law 8vor  2  50 

*  Treatise  on  the  Military  Law  of  United  States Svo,  7  oo 

Sheep,  7  50 

De  Brack's  Cavalry  Outposts  Duties.     (Carr.) 24010,  morocco,  2  oo 

Dietz's  Soldier's  First  Aid  Handbook i6mo,  morocco,  i  25 

*  Dredge's  Modern  French  Artillery 4to,  half  morocco,  15  oo 

Durand's  Resistance  and  Propulsion  of  Ships 8vo,  5  co 

*  Dyer's  Handbook  of  Light  Artillery i2mo,  3  oo 

Eissler's  Modern  High  Explosives 8vo,  4  oo 

*  Fiebeger's  Text-book  on  Field  Fortification Small  Svo,  2  co 

Hamilton's  The  Gunner's  Catechism iSmo,  i  oo 

*  Hoff's  Elementary  Naval  Tactics Svo,  i  50 

Ingalls's  Handbook  of  Problems  in  Direct  Fire Svo,  4  oo 

*  Ballistic  Tables Svo,  i  50 

*  Lyons's  Treatise  on  Electromagnetic  Phenomena.   Vols.  I.  and  II. .  Svo,  each,  6  oo 

*  Mahan's  Permanent  Fortifications.     (Mercur.) Svo,  half  morocco,  7  50 

Manual  for  Courts-martial i6mo,  morocco,  i  50 

*  Mercur's  Attack  of  Fortified  Places i2mo,  2  oo 

*  Elements  of  the  Art  of  War Svo,  4  oo 

Metcalf's  Cost  of  Manufactures — And  the  Administration  of  Workshops.  .Svo,  5  oo 

*  Ordnance  and  Gunnery.     2  vols i2mo,  5  oo 

Murray's  Infantry  Drill  Regulations iSmo,  paper,  10 

Nixon's  Adjutants'  Manual 24010,  i  oo 

Peabody's  Naval  Architecture Svo,  7  $o> 

2 


*  Phelps's  Practical  Marine  Surveying 8vo,  2  50 

Powell's  Army  Officer's  Examiner I2mo,  4  oo 

Sharpe's  Art  of  Subsisting  Armies  in  War i8mo,  morocco,  i  50 

*  Walke's  Lectures  on  Explosives 8vo,  4  oo 

*  Wheeler's  Siege  Operations  and  Military  Mining 8vo,  2  oo 

Winthrop's  Abridgment  of  Military  Law i2mo,  2  50 

Woodhull's  Notes  on  Military  Hygiene \. i6mo,  i  50 

Young's  Simple  Elements  of  Navigation i6mo,  morocco,  i  oo 

Second  Edition,  Enlarged  and  Revised i6mo,  morocco,  2  oo. 

ASSAYING. 
Fletcher's  Practical  Instructions  in  Quantitative  Assaying  with  the  Blowpipe. 

i2mo,  morocco,  i  50* 

Furman's  Manual  of  Practical  Assaying 8vo,  3  oo 

Lodge's  Notes  on  Assaying  and  Metallurgical  Laboratory  Experiments.  .  .  .8vo,  3  oo 

Miller's  Manual  of  Assaying I2mo,  i  oo 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores 8vo,  2  oo 

Ricketts  and  Miller's  Notes  on  Assaying 8vo,  3  oo 

Ulke's  Modern  Electrolytic  Copper  Refining ? 8vo,  3  oo 

Wilson's  Cyanide  Processes i2mo,  i  50 

Chlorination  Process i2mo,  i  50 

ASTRONOMY. 

Comstock's  Field  Astronomy  for  Engineers 8vo,  2  50- 

Craig's  Azimuth 4to,  3  50 

Doolittle's  Treatise  on  Practical  Astronomy 8vo,  4  oo 

Gore's  Elements  of  Geodesy 8vo,  2  50 

Hayford's  Text-book  of  Geodetic  Astronomy 8vo,  3  oo- 

Merriman's  Elements  of  Precise  Surveying  and  Geodesy 8vo,  2  '50 

*  Michie  and  Harlow's  Practical  Astronomy 8vo,  3  oo 

*  White's  Elements  of  Theoretical  and  Descriptive  Astronomy i2mo,  2  oa 

BOTANY. 

Davenport's  Statistical  Methods,  with  Special  Reference  to  Biological  Variation. 

i6mo,  morocco,    i  25 

Thome'  and  Bennett's  Structural  and  Physiological  Botany i6mo,    2  25 

Westermaier's  Compendium  of  General  Botany.     (Schneider.) 8vo,    2  oo 

CHEMISTRY. 

Adriance's  Laboratory  Calculations  and  Specific  Gravity  Tables I2mo,  I  25 

Allen's  Tables  for  Iron  Analysis 8vo,  3  oo 

Arnold's  Compendium  of  Chemistry.     (Mandel.) Small  8vo,  3  50 

Austen's  Notes  for  Chemical  Students 12010,  I  50 

Bernadou's  Smokeless  Powder. — Nitro-cellulose,  and  Theory  of  the  Cellulose 

Molecule 12010,  2  50 

Bolton's  Quantitative  Analysis 8vo ,  i  50- 

*  Browning's  Introduction  to  the  Rarer  Elements 8vo,  i  50- 

Brush  and  Penfield's  Manual  of  Determinative  Mineralogy 8vo,  4  oo 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.    (Boltwood. ).  .8vo,  3  oo 

Cohn's  Indicators  and  Test-papers i2mo,  2  oo- 

Tests  and  Reagents 8vo,  3  oo 

Crafts's  Short  Course  in  Qualitative  Chemical  Analysis.   (Schaeffer.).  .  .12010,  i  sa 
Dolezalek's  Theory  of  the  Lead  Accumulator   (Storage  Battery).        (Von 

Ende.) i2mo,  2  50 

Drechsel's  Chemical  Reactions.     (Merrill.) i2mo,  i  25 

Duhem's  Thermodynamics  and  Chemistry.     (Burgess.) ,  .8vo,  4  oo 

Eissler's  Modern  High  Explosives 8vo,  4  oo 

Effront's  Enzymes  and  their  Applications.     (Prescott.) 8vo,  3  oo 

Erdmann's  Introduction  to  Chemical  Preparations.     (Dunlap.) i2mo,  i  25". 

3 


Fletcher's  Practical  Instructions  in  Quantitative  Assay irg  with  the  Blowpipe. 

i2mo,  morocco,     i  50 

Fowler's  Sewage  Works  Analyses i2mo,    2  oo 

Fresenius's  Manual  of  Qualitative  Chemical  Analysis.     (Wells.) 8vo,    5  oo 

Manual  of  Qualitative  Chemical  Analysis.  Part  I.  Descriptive.  (Weils.)  8vo,     3  oo 
System   of    Instruction    in    Quantitative    Chemical    Analysis.       (Coin.) 

2  vols 8vo,  12  50 

Fuertes's  Water  and  Public  Health 12010,     i  50 

Furman's  Manual  of  Practical  Assaying 8vo,    3  oo 

*  Getman's  Exercises  in  Physical  Chemistry i2mo,     2  oo 

Gill's  Gas  and  Fuel  Analysis  for  Engineers i2mo,     i   25 

Groten/eit's  Principles  of  Modern  Dairy  Practice.     (Woll.) i2mo,    2  oo 

Hammarsten's  Text-book  of  Physiological  Chemistry.     (Mandel.) 8vo,     4  oo 

Helm's  Principles  of  Mathematical  Chemistry.      (Morgan.) i2iro,     i   50 

Hering's  Ready  Reference  Tables  (Conversion  Factors) i6rr.o,  rr.orocco,    2  50 

Hind's  Inorganic  Chemistry 8vo,    3  oo 

Laboratory  Manual  for  Students i2mo,         75 

Holleman's  Text-book  of  Inorganic  Chemistry.     (Cooper.) 8vo,     2  50 

Text-book  of  Organic  Chemistry.     (Walker  and  Mott.) 8vo,    2  50 

*  Laboratory  Manual  of  Organic  Chemistry.     (Walker.) i2mo,     i  oo 

Hopkins's  Oil-chemists'  Handbook 8vo,    3  oo 

Jackson's  Directions  for  Laboratory  Work  in  Physiological  Chen.Isiry.  .8vo,     i   23 

Keep's  Cast  Iron 8vo,    2  50 

Ladd's  Manual  of  Quantitative  Chemical  Analysis i2mo,     i  oo 

Landauer's  Spectrum  Analysis.     (Tingle.) 8vo,    3  oo 

*  Langworthy   and   Austen.         The   Occurrence   of  Aluminium   in   Vege'able 

Products,  Animal  Products,  and  Natural  Waters 8vo,  2  oo 

Lassar-Cohn's  Practical  Urinary  Analysis.  (Lorenz.) i2mo,  i  oo 

Application  of  Some  General  Reactions  to  Investigations  in  Organic 

Chemistry.  (Tingle.) i2mo,  i  oo 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control. 8vo,  7  50 

Lob's  Electrolysis  and  Electrosynthesis  of  Organic  Compounds.  (Lorenz. ).i2mo,  i  oo 
Lodge's  Notes  on  Assaying  and  Metallurgical  Laboratory  Experiments.  ..  .8vo,  3  co 

Lunge's  Techno-chemical  Analysis.  (Cohn.) i2mo,  i  oo 

Mandel's  Handbook  for  Bio-chemical  Laboratory i2mo,  i  50 

*  Martin's  Laboratory  Guide  to  Qualitative  Analysis  with  the  Blowpipe .  .  i2mo,        Co 
Mason's  Water-supply.     (Considered  Principally  from  a  Sanitary  Standpoint.) 

3d  Edition,  Rewritten 8vo,    4  oo 

Examination  of  Water.     (Chemical  and  Bacteriological.) i2rro,     i   25 

Matthew's  The  Textile  Fibres 8vo,    3  50 

Meyer's  Determination  of  Radicles  in  Carbon  Compounds.     (Tingle.).  .12010,     i  oo 

Miller's  Manual  of  Assaying i2mo,     i  .00 

Mixter's  Elementary  Text-book  of  Chemistry i2mo,     i  51 

Morgan's  Outline  of  Theory  of  Solution  and  its  Results i2iro,     i  oc 

Elements  of  Physical  Chemistry i2mo,     2  oo 

Morse's  Calculations  used  in  Cane-sugar  Factories i6mo,  morocco,     i  50 

Mulliken's  General  Method  for  the  Identification  of  Pure  Organic  Compounds. 

Vol.  I ' Large  8vo,    5  oo 

O'Brine's  Laboratory  Guide  in  Chemical  Analysis 8vo,    2  oo 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores 8vo,    2  oo 

Ostwald's  Conversations  on  Chemistry.     Part  One.     (Ramsey.) 12010,     i  50 

Ostwald's  Conversations  on  Chemistry.     Part  Two.     (Turnbull.).     (In  Press.) 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mireral  Tests. 

8vo,  paper,         50 

Pictet's  The  Alkaloids  and  their  Chemical  Constitution.     (Biddle.) 8vo,    5  oo 

Pinner's  Introduction  to  Organic  Chemistry.     (Austen.) i2mo,     i  50 

Poole's  Calorific  Power  of  Fuels 8vo,    3  oo 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis i2mo,    i  25 

4 


*  Relslg's  Guide  to  Piece-dyeing 8vo,  25  oo 

Richards  and  Woodman's  Air,  Water,  and  Food  from  a  Sanitary  Standpoint  8vo,  2  oo 

Richards's  Cost  of  Living  as  Modified  by  Sanitary  Science i2mo,  i  oo 

Cost  of  Food,  a  Study  in  Dietaries i2mo,  i  oo 

*  Richards  and  Williams's  The  Dietary  Computer 8vo,  i  50 

Ricketts  and  Russell's  Skeleton  Notes  upon  Inorganic   Chemistry.      (Part  I. 

Non-metallic  Elements.) 8vo,  morocco,  75 

Ricketts  and  Miller's  Notes  on  Assaying.  .  .  . ' 8vo,  3  oo 

Rideal's  Sewage  and  the  Bacterial  Purificat  on  of  Sewage £vo,  3  50 

Disinfection  and  the  Preservation  of  Food 8vo,  4  oo 

Rigg's  Elementary  Manual  for  the  Chemical  Laboratory 8vo,  i  25 

Rostoski's  Serum  Diagnosis.  (Bolduan.) i2rno,  i  oo 

Ruddiman's  Incompatibilities  in  Prescriptions .  £vo,  2  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish .  .Svo,  3  oo 

Salkowski's  Physiological  and  Pathological  Chemistry.  (Orndorff.) Svo,  2  50 

Schimpf's  Text-book  of  Volumetric  Analysis i2mo,  2  50 

Essentials  of  Volumetric  Analysis i2mo,  i  25 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6mo,  morocco,  3  oo 

Handbook  for  Sugar  Manufacturers  and  their  Chemists.  .  i6mo,  morocco,  2  oo 

Stockbridge's  Rocks  and  Soils Svo,  2  50 

*  Tillman's  Elementary  Lessons  in  Heat Svo,  i  50 

Descriptive  General  Chemistry Svo,  3  oo 

Treadwell's  Qualitative  Analysis.  (Hall.) Svo,  3  oo 

Quantitative  Analysis.  (Hall.) Svo,  4  oo 

lurneaure  and  Russell's  Public  Water-supplies Svo,  5  oo 

Van  Deventer's  Physical  Chemistry  for  Beginners.  (Boltwood.) i2rno,  i  50 

*  Walke's  Lectures  on  Explosives Svo,  4  oo 

Washington's  Manual  of  the  Chemical  Analysis  of  Rocks Svo,  2  oo 

Wassermann's  Immune  Sera :  Haemolyslns,  Cytotoxins,  and  Precipitir.s.    (Bol- 
duan.)   i2mo,  i  oo 

Well's  Laboratory  Guide  in  Qualitative  Chemical  Analysis Svo,  i  50 

Short  Course  in  inorganic  Qualitative  Chemical  Analysis  for  Engineering 

Students i2mo,  i   50 

Text-book  of  Chemical  Arithmetic.     (In  press.). 

Whipple's  Microscopy  of  Drinking-water Svo,  3  50 

Wilson's  Cyanide  Processes i2mo,  i  50 

Chlorination  Process i2mo,  i   50 

Wulling's    Elementary    Course    in  Inorganic,  Pharmaceutical,  and  Medical 

Chemistry i2mo,  2  oo 

CIVIL  ENGINEERING. 

BRIDGES    AND    ROOFS.       HYDRAULICS.       MATERIALS    OF    ENGINEERING. 
RAILWAY  ENGINEERING. 

Baker's  Engineers'  Surveying  Instruments 12 mo,  3  oo 

Bixby's  Graphical  Computing  Table Paper  19^X24!  inches.  25 

**  Burr's  Ancient  and  Modern  Engineering  and  the  Isthmian  Canal.     (Postage, 

27  cents  additional.) Svo,  3  50 

Comstock's  Field  Astronomy  for  Engineers Svo,  2  50 

Davis's  Elevation  and  Stadia  Tables Svo,  i  oo 

Elliott's  Engineering  for  Land  Drainage I2mo,  i  50 

Practical  Farm  Drainage i2mo,  i  oo 

Fiebeger's  Treatise  on  Civil  Engineering.     (In  press.) 

Folwell's  Sewerage.     (Designing  and  Maintenance.) Svo,  3  oo 

Freitag's  Architectural  Engineering.     2d  Edition,  Rewritten Svo,  3  50 

French  and  Ives's  Stereotomy Svo,  2  50 

Goodhue's  Municipal  Improvements . .  •  •  •  larno,  i  75 

Goodrich's  Economic  Disposal  of  Towns'  Refuse Svo,  3  50 

Gore's  Elements  of  Geodesy Svo,  2  50 

Hayford's  Text-book  of  Geodetic  Astronomy Svo,  3  oo 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

5 


"Howe's  Retaining  Walls  for  Earth i2mo,  i   25 

Johnson's  (J.  B.)  Theory  and  Practice  of  Surveying Small  8vo,  4  oo 

Johnson's  (L.  J.)  Statics  by  Algebraic  and  Graphic  Methods 8vo,  2  oo 

Laplace's  Philosophical  Essay  on  Probabilities.    (Truscott  and  Emory.) .  i2mo,  2  oo 

Mahan's  Treatise  on  Civil  Engineering.     (1873.)     (Wood.) 8vo,  5  oo 

*  Descriptive  Geometry 8vo,  i  50 

Merriman's  Elements  of  Precise  Surveying  and  Geodesy .8vo,  2  50 

Elements  of  Sanitary  Engineering 8vo,  2  oo 

Merriman  and  Brooks's  Handbook  for  Surveyors i6mo,  morocco,  2  oo 

Nugent's  Plane  Surveying 8vo,  3  50 

•Ogden's  Sewer  Design i2mo,  2  oo 

Patton's  Treatise  on  Civil  Engineering 8vo  half  leather,  7  50 

Reed's  Topographical  Drawing  and  Sketching 4to,  5  oo 

Rideal's  Sewage  and  the  Bacterial  Purification  of  Sewage 8vo,  3  50 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry 8vo,  i  50 

Smith's  Manual  of  Topographical  Drawing.     (McMillan.) 8vo,  2  50 

Sondericker's  Graphic  Statics,  with  Applications  to  Trusses,  Beams,  and  Arches. 

8vp,  2  oo 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  oo 

*  Trautwine's  Civil  Engineer's  Pocket-book i6mo,  morocco,  5  oo 

Wait's  Engineering  and  Archi'. ectural  Jurisprudence 8vo,  6  oo 

Sheep,  6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo,  5  oo 

Sheep,  5  50 

Law  of  Contracts 8vo,  3  oo 

^Warren's  Stereotomy — Problems  in  Stone-cutting 8vo,  2  50 

Webb's  Problems  in  the  Use  and  Adjustment  of  Engineering  Instruments. 

i6mo,  morocco,  i   25 

*  Wheeler's  Elementary  Course  of  Civil  Engineering 8vo,  4  oo 

Wilson's  Topographic  Surveying 8vo,  3  50 

BRIDGES  ATO   ROOFS. 

Boiler's  Practical  Treatise  on  the  Construction  of  Iron  Highway  Bridges.  .8vo,  2  oo 

*  Thames  River  Bridge .  .  4to,  paper,  5  oo 

Burr's  Course  on  the  Stresses  in  Bridges  and  Roof  Trusses,  Arched  Ribs,  and 

Suspension  Bridges 8vo,  3  50 

Burr  and  Talk's  Influence  Lines  for  Bridge  and  Roof  Computations.  .  .  .8vo,  3  oo 

Du  Bois's  Mechanics  of  Engineering.     Vol.  II Small  4to,  10  oo 

Poster's  Treatise  on  Wooden  Trestle  Bridges 4to,  5  oo 

Fowler's  Ordinary  Foundations 8vo,  3  50 

Greene's  Roof  Trusses 8vo,  i   25 

Bridge  Trusses 8vo,  2  50 

Arches  in  Wood,  Iron,  and  Stone 8vo,  2  50 

JHowe's  Treatise  on  Arches 8vo,  4  oo 

Design  of  Simple  Roof-trusses  in  Wood  and  Steel 8vo,  2  oo 

Johnson,  Bryan,  and  Turneaure's  Theory  and  Practice  in  the  Cesiprirg  of 

Modern  Framed  Structures Small  4to,  10  oo 

Merriman  and  Jacoby's  Text-book  on  Roofs  and  Bridges: 

Part  I.     Stresses  in  Simple  Trusses ' 8vo,  2  50 

Part  II.     Graphic  Statics .   8vo,  2  50 

Part  III.     Bridge  Design -  -  8vo,  2  50 

Part  IV.     Higher  Structures -    8vo,  2  50 

Morison's  Memphis  Bridge % •   4to,  10  oo 

Waddell's  De  Pontibus,  a  Pocket-book  for  Bridge  Engineers.  .  i6mo,  morocco,  3  oo 

Specifications  for  Steel  Bridges i2mo,  i   25 

Wood's  Treatise  on  the  Theory  of  the  Construction  of  Bridges  and  Roofs .  .  8vo,  2  oo 
Wright's  Designing  of  Draw-spans : 

Part  I.     Plate-girder  Draws 8vo,  2  50 

Part  II.     Riveted-truss  and  Pin-connected  Long-span  Draws 8vo,  2  50 

Two  parts  in  one  volume 8vo,  3  5° 


HYDRAULICS. 

Bazin's  Experiments  upon  the  Contraction  of  the  Liquid  Vein  Issuing  from 

an  Orifice.     (Trautwine.) 8vo,  2  oo 

Bovey's  Treatise  on  Hydraulics 8vo,  5  oo 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

Diagrams  of  Mean  Velocity  of  Water  in  Open  Channels payer,  i  50 

Coffin's  Graphical  Solution  of  Hydraulic  Problems i6mo,  morocco,  2  50 

Flather's  Dynamometers,  and  the  Measurement  of  Power I2mo,  3  oo 

FolwelTs  Water-supply  Engineering 8vo,  4  oo 

Frizell's  Water-power .  .  .8vo,  5  oo 

Fuertes's  Water  and  Public  Health .  .  i2mo,  i  50 

Water-filtration  Works . .  i2mo,  2  50 

Ganguillet  and  Kutter's  General  Formula  for  the  Uniform  Flow  of  Water  in 

Rivers  and  Other  Channels.     (Bering  and  Trautwine.) 8vo  4  oo 

Hazen's  Filtration  of  Public  Water-supply 8vo,  3  oo 

Hazlehurst's  Towers  and  Tanks  for  Water-works 8vo,  2  50 

Herschel's  115  Experiments  on  the  Carrying  Capacity  of  Large,  Piveted,  Metal 

Conduits .  .  .  8vo,  2  oo 

Mason's  Water-supply.     (Considered  Principally  from  a  Sanitary  Standpoint.) 

8vo,  4  oo 

Merriman's  Treatise  on  Hydraulics 8vo,  5  oo 

*  Michie's  Elements  of  Analytical  Mechanics .  .8vo,  4  oo 

Schuyler's   Reservoirs   for   Irrigation,    Water-power,   and   Domestic    Water- 
supply L&i.^e  8vo,  5  oo 

**  Thomas  and  Watt's  Improvement  of  Rivers.     (Post.,  440.  additional.)  4to,  6  oo 

Turneaure  and  Russell's  Public  Water-supplies ?vo,  5  oo 

Wegmann's  Design  and  Construction  of  Dams 4to,  5  oo 

Water-supply  of  the  City  of  New  York  from  1658  to  1895 4to,  10  oo 

Wilson's  Irrigation  Engineering .  .Small  8vo,  4  oo 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  oo 

Wood's  Turbines 8vo,  2  50 

Elements  of  Analytical  Mechanics 8vo,  3  oo 

MATERIALS  OF  ENGINEERING. 

Baker's  Treatise  on  Masonry  Construction .  .8vo,  5  oo 

Roads  and  Pavements .    3vo,  5  oo 

Black's  United  States  Public  Works Ob?ong  4*0,  5  oo 

Bovey's  Strength  of  Materials  and  Theory  of  Structures Svo.  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering 8vo.  7  50 

Byrne's  Highway  Construction 8vo,  5  oo 

Inspection  of  the  Materials  and  Workmanship  Employed  in  Construction. 

i6mo,  3  oo 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

Du  Bois's  Mechanics  of  Engineering.     Vol.  I Small  4to,  7  50 

Johnson's  Materials  of  Construction Large  8vo,  6  oo 

Fowler's  Ordinary  Foundations 8vo,  3  50 

Keep's  Cast  Iron 8vo,  2  50 

Lanza's  Applied  Mechanics 8vo,  7  50 

Marten's  Handbook  on  Testing  Materials.     (Henning.)     2  vols 8vo,  7  50 

Merrill's  Stones  for  Building  and  Decoration 8vo,  5  oo 

Merriman's  Text-book  on  the  Mechanics  of  Materials 8vo,  4  oo 

Strength  of  Materials i2mo,  i  oo 

Metcalf's  Steel.     A  Manual  for  Steel-users ."  .  .  i2mo,  2  oo 

Patton's  Practical  Treatise  on  Foundations 8vo,  5  oo 

Richardson's  Modern  Asphalt  Pavements.     (In  press.) 

Richey's  Handbook  for  Superintendents  of  Construction i6mo,  mor.,  4  oo 

Rockwell's  Roads  and  Pavements  in  France i2mo,  i  23 

7 


Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  oo 

Smith's  Materials  of  Machines i2mo,  i  oo 

Snow's  Principal  Species  of  Wood 8vo,  3  50 

Spalding's  Hydraulic  Cement .  i2mo,  2  oo 

Text-book  on  Roads  and  Pavements i2mo,  2  oo 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  oo 

Thurston's  Materials  of  Engineering.     3  Parts ' .  ,8vo,  8  oo 

Part  I.     Non-metallic  Materials  of  Engineering  and  Ivletailurcy 8\o,  2  oo 

Part  II.     Iron  and  Steel 8vo,  3  50 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

Thurston's  Text-book  of  the  Materials  of  Construction 8vo,  5  oo 

Tillson's  Street  Pavements  and  Paving  Materials 8vo,  4  oo 

Waddell's  De  Pontibus.    (A  Pocket-book  for  Bridge  Engineers.).  .  i6mo,  mor.,  3  oo 

Specifications  for  Steel  Bridges i2mo,  i  25 

Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials,  and  an  Appendix  on 

the  Preservation  of  Timber 8vo,  2  oo 

Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo,  3  oo 

Wood's  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

Steel .  . 8vo,  4  oo 

RAILWAY  ENGINEERING. 

Andrew's  Handbook  for  Street  Railway  Engineers 3x5  inches,  morocco,  i  25 

Berg's  Buildings  and  Structures  of  American  Railroads 4to,  5  oo 

Brook's  Handbook  of  Street  Railroad  Location >.  .  .  i6mo,  morocco,  i  50 

Butt's  Civil  Engineer's  Field-book i6mo,  morocco,  2  50 

Crandall's  Transition  Curve i6mo,  morocco,  i  50 

Railway  and  Other  Earthwork  Tables 8vo,  i  50 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book    .  i6mo,  morocco,  5  oo 

Dredge's  History  of  the  Pennsylvania  Railroad:    (1879) Paper,  5  oo 

*  Drinker's  Tunnelling,  Explosive  Compounds,  and  Rock  Drills. 4to,  half  mor.,  25  oo 

Fisher's  Table  of  Cubic  Yards Cardboard,  25 

Godwin's  Railroad  Engineers'  Field-book  and  Explorers'  Guide.      i6mo,  mor.,  2  50 

Howard's  Transition  Curve  Field-book i6mo,  morocco,  i  50 

Hudson's  Tables  for  Calculating  the  Cubic  Contents  of  Excavations  and  Em- 
bankments  8vo,  i  oo 

Molitor  and  Beard's  Manual  for  Resident  Engineers i6mo,  i  oo 

Nagle's  Field  Manual  for  Railroad  Engineers i6mo,  morocco,  3  oo 

Philbrick's  Field  Manual  for  Engineers i6mo,  morocco,  3  oo 

Searles's  Field  Engineering i6mo,  morocco,  3  oo 

Railroad  Spiral i6mo,  morocco,  i  50 

Taylor's  Prismoidal  Formulae  and  Earthwork 8vo,  i  50 

*  Trautwine's  Method  of  Calculating  the  Cube  Contents  of  Excavations  and 

Embankments  by  the  Aid  of  Diagrams 8vo,  2  oo 

The  Field  Practice  of  Laying  Out  Circular  Curves  for  Railroads. 

i2mo,  morocco,  2  50 

Cross-section  Sheet Paper,  25 

Webb's  Railroad  Construction i6mo,  morocco,  5  oo 

Wellington's  Economic  Theory  of  the  Location  of  Railways Small  8vo,  5  oo. 

DRAWING. 

Barr's  Kinematics  of  Machinery 8vo,     2  50 

*  Bartlett's  Mechanical  Drawing 8vo,     3  oo 

*  "                     "                     "         Abridged  Ed 8vo,  i   50 

Coolidge's  Manual  of  Drawing 8vo,  paper  i  oo 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  Engi- 
neers  '. Oblong  4to,  2  50 

Durley's  Kinematics  of  Machines 8vo,     4  oo 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications 8vo.    2  50 

8 


Hill's  Text-book  on  Shades  and  Shadows,  and  Perspective 8vo,  2  oo 

Jamison's  Elements  of  Mechanical  Drawing 8vo,  2  50 

Jones's  Machine  Design: 

Part  I.     Kinematics  of  Machinery 8vo,  i  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

MacCord's  Elements  of  Descriptive  Geometry 8vo,  3  oo 

Kinematics;  or,  Practical  Mechanism 8vo,  5  oo 

Mechanical  Drawing 4to,  4  oo 

Velocity  Diagrams. 8vo,  i   50 

*  Mahan's  Descriptive  Geometry  and  Stone-cutting 8vo,  i   50 

Industrial  Drawing.     (Thompson.) 8vo,  3  50 

Moyer's  Descriptive  Geometry.     (In  press.) 

Reed's  Topographical  Drawing  and  Sketching 4to,  5  oo 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  oo 

Robinson's  Principles  of  Mechanism 8vo,  3  oo 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  oo 

Smith's  Manual  of  Topographical  Drawing.     (McMillan.) 8vo,  2  50 

Warren's  Elements  of  Plane  and  Solid  Free-hand  Geometrical  Drawing.  ?2mo, 


Drafting  Instruments  and  Operations i2mo, 

Manual  of  Elementary  Projection  Drawing i2mo, 

Manual  of  Elementary  Problems  in  the  Linear  Perspective  of  Form  and 

Shadow i2mo, 

Plane  Problems  in  Elementary  Geometry i2mo, 


oo 
25 

Primary  Geometry i2mo,  75 

Elements  of  Descriptive  Geometry,  Shadows,  and  Perspective 8vo,  3  50 

General  Problems  of  Shades  and  Shadows 8vo,  3  oo 

Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Problems,  Theorems,  and  Examples  in  Descriptive  Geometry 8vo,  2  50 

Weisbach's  Kinematics  and  Powerof  Transmission.    (Hermann  and  Klein)8vo,  5  oo 

Whelpley's  Practical  Instruction  in  the  Art  of  Letter  Engraving i2mo,  2  oo 

Wilson's  (H.  M.)  Topographic  Surveying 8vo,  3  50 

Wilson's  (V.  T.)  Free-hand  Perspective 8vo,  2  50 

Wilson's  (V.  T.)  Free-hand  Lettering , 8vo,  i  oo 

Woolf's  Elementary  Course  in  Descriptive  Geometry Large  8vo,  3  OQ 


ELECTRICITY  AND  PHYSICS. 

Anthony  and  Brackett's  Text-book  of  Physics.     (Magie.) Small  8vo,  3  oo 

Anthony's  Lecture-notes  on  the  Theory  of  Electrical  Measurements.  .  .  .  i2mo,  i   oo 

Benjamin's  History  of  Electricity , 8vo,  3  oo 

Voltaic  Cell 8vo,  3  oo 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.     (Boltwood.).8vo,  3  oo 

Crehore  and  Squier's  Polarizing  Photo-chronograph • 8vo,  3  oo 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  i6mo,  morocco,  5  oo 
Dolezalek's    Theory   of    the    Lead   Accumulator    (Storage    Battery).      (Von 

Ende.) i2mo,  50 

Duhem's  Thermodynamics  and  Chemistry.     (Burgess.) 8vo,  oo 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  oo 

Gilbert's  De  Magnete.     (Mottelay.) 8vo,  50 

Hanchett's  Alternating  Currents  Explained i2mo,  oo 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  50 

Holman's  Precision  of  Measurements 8vo,  oo 

Telescopic   Mirror-scale  Method,  Adjustments,  and   Tests.  .  .  .Large  8vo,  75 

Kinzbrunner's  Testing  of  Continuous-Current  Machines 8vo.  2  oo 

Landauer's  Spectrum  Analysis.     (Tingle.) 8vo,  3  oo 

Le  Chatelien's  High-temperature  Measurements.  (Boudouard — Burgess.)  I2mo,  3  oo 

Lob's  Electrolysis  and  Electrosynthesis  of  Organic  Compounds.  (Lorenz.)  i2mo,  i  oo 

9 


*  Lyons's  Treatise  on  Electromagnetic  Phenomena.   Vols.  I.  and  II.  8vo,  each,  6  oo 

*  Michie's  Elements  of  Wave  Motion  Relating  to  Sound  and  Light 8vo,  4  oo 

Niaudet's  Elementary  Treatise  on  Electric  Batteries.     (Fishback.) i2mo»  2  50 

*  Rosenberg's  Electrical  Engineering.     (Haldane  Gee — Kinzbrunner.).  . .8vo,  i  50 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     Vol.  1 8vo,  2  50 

Thurston's  Stationary  Steam-engines 8vo,  2  50 

•*  Tillman's  Elementary  Lessons  in  Heat .  .8vo,  i  50 

Tory  and  Pitcher's  Manual  of  Laboratory  Physics Small  8vo,  2  oo 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  3  oo 

LAW. 

*  Davis's  Elements  of  Law 8vo,  2  50 

*  Treatise  on  the  Military  Law  of  United  States 8vo,  7  oo 

Sheep,  7  50 

Manual  for  Courts-martial.  . i6mo,  morocco,  i  50 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,  6  oo 

Sheep,  6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo,  5  oo 

Sheep,  5  50 

Law  of  Contracts 8vo,  3  oo 

"Wlnthrop's  Abridgment  of  Military  Law i2mo,  2  50 

MANUFACTURES. 

Bernadou's  Smokeless  Powder — Nitro-cellulose  and  Theory  of  the  Cellulose 

Molecule i2mo,  2  50 

Bolland's  Iron  Founder i2mo,  2  50 

"  The  Iron  Founder,"  Supplement i2mo,  2  50 

Encyclopedia  of  Founding  and  Dictionary  of  Foundry  Terms  Used  in  the 

Practice  of  Moulding i2mo,  3  oo 

Eissler's  Modern  High  Explosives 8vo,  4  oo 

Eff rent's  Enzymes  and  their  Applications.     (Prescott.) 8vo,  3  oo 

Fitzgerald's  Boston  Machinist i2mo,  i  oo 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,  i  oo 

Hopkin's  Oil-chemists'  Handbook 8vo,  3  oo 

Keep's  Cast  Iron 8vo,  2  50 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control Large  8vo,  7  50 

Matthews's  The  Textile  Fibres 8vo,  3  50 

Metcalf's  Steel.     A  Manual  for  Steel-users i2mo,  2  oo 

Metcalfe's  Cost  of  Manufactures — And  the  Administration  of  Workshops. 8vo,  5  oo 

Meyer's  Modern  Locomotive  Construction 4to,  10  oo 

Morse's  Calculations  used  in  Cane-sugar  Factories i6mo,  morocco,  i  50 

*  Reisig's  Guide  to  Piece-dyeing 8vo,  25  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  oo 

Smith's  Press-working  of  Metals 8vo,  3  oo 

Spalding's  Hydraulic  Cement i2mo,  2  oo 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses.     ...  i6mo,  morocco,  3  oo 

Handbook  for  Sugar  Manufacturers  and  their  Chemists    .  i6mo,  morocco,  2  oo 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  oo 

Thurston's  Manual  of  Steam-boilers,  their  Designs,  Construction  and  Opera- 
tion  8vo,  5  oo 

*  Walke's  Lectures  on  Explosives 8vo,  4  oo 

Ware's  Manufacture  of  Sugar.     (In  press.) 

West's  American  Foundry  Practice i2mo,  2  50 

Moulder's  Text-book i2mo,  2  50 

10 


Wolff's  Windmill  as  a  Prime  Mover 8vo,    3  oo 

Wood's  Rustless  Coatings:   Corrosion  and  Electrolysis  of  Iron  and  Steel.  .8vo,    4  »o 


MATHEMATICS. 

Baker's  Elliptic  Functions 8vo,  i  50 

*  Bass's  Elements  of  Differential  Calculus i2mo,  4  oo 

Briggs's  Elements  of  Plane  Analytic  Geometry i2ino,  oo 

Compton's  Manual  of  Logarithmic  Computations i2mo,  50 

Davis's  Introduction  to  the  Logic  of  Algebra 8vo,  50 

*  Dickson's  College  Algebra Large  i2mo,  50 

*  Introduction  to  the  Theory  of  Algebraic  Equations Large  i2mo,  25 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications 8vo,  50 

Halsted's  Elements  of  Geometry 8vo,  75 

Elementary  Synthetic  Geometry 8vo,  50 

Rational  Geometry i2mo,  75 

*  Johnson's  (J.  B.)  Three-place  Logarithmic  Tables:    Vest-pocket  size. paper,  15 

100  copies  for  5  oo 

*  Mounted  on  heavy  cardboard,  8  X  TO  inches,  25 

10  copies  for  2  oo 

Johnson's  (W.  W.)  Elementary  Treatise  on  Differential  Calculus.  . Small  8vo,  3  oo 

Johnson's  (W.  W.)  Elementary  Treatise  on  the  Integral  Calculus. Small  8vo,  i  50 

Johnson's  (W.  W.)  Curve  Tracing  in  Cartesian  Co-ordinates I2mo,  i  oo 

Johnson's  (W.  W.)  Treatise  on  Ordinary  and  Partial  Differential  Equations. 

Small  8vo,  3  50 

Johnson's  (W.  W.)  Theory  of  Errors  and  the  Method  of  Least  Squares.  i2mo,  i  50 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,  3  oo 

Laplace's  Philosophical  Essay  on  Probabilities.     (Truscott  and  Emory.).  i2mo,  2  oo 

*  Ludlow  and  Bass.     Elements  of  Trigonometry  and  Logarithmic  and  Other 

Tables 8vo,  3  oo 

Trigonometry  and  Tables  published  separately Each,  2  oo 

*  Ludlow's  Logarithmic  and  Trigonometric  Tables 8vo,  i  oo 

Maurer's  Technical  Mechanics fev , ,  4  oo 

Merriman  and  Woodward's  Higher  Mathematics 8vo,  5  oo 

Merriman's  Method  of  Least  Squares 8vo,  2  oo 

Rice  and  Johnson's  Elementary  Treatise  on  the  Differential  Calculus. .  Sm.  8vo,  3  oo 

Differential  and  Integral  Calculus.     2  vols.  in  one Small  8vo,  2  50 

Wood's  Elements  of  Co-ordinate  Geometry 8vo,  2  oo 

Trigonometry:   Analytical,  Plane,  and  Spherical i2mo,  i  oo 


MECHANICAL  ENGINEERING. 

MATERIALS.  OF  ENGINEERING,  STEAM-ENGINES  AND  BOILERS. 

Bacon's  Forge  Practice i2mo,  i  50 

Baldwin's  Steam  Heating  for  Buildings i2mo,  2  50 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

*  Bartlett's  Mechanical  Drawing 8vo,  3  oo 

*  "  "  "        Abridged  Ed 8vo,     i  50 

Benjamin's  Wrinkles  and  Recipes i2mo,     2  oo 

Carpenter's  Experimental  Engineering 8vo,    6  oo 

Heating  and  Ventilating  Buildings 8vo,    4  oo 

Cary's  Smoke  Suppression  in  Plants  using  Bituminous  Coal.     (In  Prepara- 
tion.) 

Clerk's  Gas  and  Oil  Engine Small  8vo,    4  oo 

Coolidge's  Manual  of  Drawing 8vo,  paper,     i  oo 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  En- 
gineers  Oblong  4to,    2  50 

11 


Cromwell's  Treatise  on  Toothed  Gearing i2mo,  i  50 

Treatise  on  Belts  and  Pulleys i2mo,  i  50 

Durley's  Kinematics  of  Machines 8vo,  4  oo 

Flather's  Dynamometers  and  the  Measurement  of  Power i2mo,  3  oo 

Rope  Driving i2mo,  2  oo 

Gill's  Gas  and  Fuel  Analysis  for  Engineers i2mo,  i  25 

Hall's  Car  Lubrication i2mo,  i  oo 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

Button's  The  Gas  Engine 8vo,  5  oo 

Jamison's  Mechanical  Drawing 8vo,  2  50 

Jones's  Machine  Design : 

Part  I.     Kinematics  of  Machinery 8vo,  i   50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

Kent's  Mechanical  Engineers'  Pocket-book i6mo,  morocco,  5  oo 

Kerr's  Power  and  Power  Transmission 8vo,  2  oo 

Leonard's  Machine  Shop,  Tools,  and  Methods.     (In  press.) 

Lorenz's  Modern  Refrigerating  Machinery.     (Pope,  Haven,  and  Dean.)     (In  press,  i 

MacCord's  Kinematics;   or,  Practical  Mechanism 8vo,  5  oo 

Mechanical  Drawing 4to,  4  oo 

Velocity  Diagrams 8vo,  i   50 

Mahan's  Industrial  Drawing.      (Thompson.) 8vo,  3  50 

Poole's  Calorific  Power  of  Fuels 8vo,  3  oo 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8 vo,  3  oo 

Richard's  Compressed  Air ' i2mo,  i   50 

Robinson's  Principles  of  Mechanism 8vo,  3  oo 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  oo 

Smith's  Press-working  of  Metals ^ 8vo,  3  oo 

Thurston's    Treatise    on    Friction  and    Lost    Work    in    Machinery   and    Mill 

Work 8vo,  3  oo 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics .  i2n:o,  i   oo 

Warren's  Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Weisbach's    Kinematics    and    the    Power    of    Transmission.      (Herrmann — 

Klein.) 8vo,  5  oo 

Machinery  of  Transmission  and  Governors.      (Herrmann — Klein.).  .8vo,  5  oo 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  oo 

Wood's  Turbines 8vo,  2  50 


MATERIALS   OF   ENGINEERING. 

Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering.     6th  Edition. 

Reset 8vo,  7  50 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

Johnson's  Materials  of  Construction 8vo,  6  oo 

Keep's  Cast  Iron 8vo,  2  50 

Lanza's  Applied  Mechanics 8vo,  7  50 

Martens's  Handbook  on  Testing  Materials.     (Henning.) 8vo,  7  50 

Merriman's  Text-book  on  the  Mechanics  of  Materials 8vo,  4  oo 

Strength  of  Materials I2mo,  i  oo 

Metcalf's  Steel.     A  manual  for  Steel-users i2mo.  2  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  oo 

Smith's  Materials  of  Machines .  .  I2mo,  i  oo 

Thurston's  Materials  of  Engineering 3  vols.,  8vo,  8  oo 

Part  II.     Iron  and  Steel 8vo,  3  50 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo>  2  5O 

Text-book  of  the  Materials  of  Construction 8vo,  5  oo. 

12 


Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials  au ."  *-n  Appendix  on 

the  Preseivation  of  Timber 8vo,    2  oo 

Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo,    3  oo 

Wood's  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

Steel 8vo,    4  oo 


STEAM-ENGINES  AND  BOILERS. 

Berry's  Temperature-entropy  Diagram i2mo,  i  25 

Carnot's  Reflections  on  the  Motive  Power  of  Heat.     (Thurston.).  ....  i2mo,  i  50 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  .  .  .i6mo,  mor.,  5  oo 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,  i  oo 

Goss's  Locomotive  Sparks , .  .8vo,  2  oo 

Hemenway's  Indicator  Practice  and  Steam-engine  Economy i2mo,  2  oo 

Button's  Mechanical  Engineering  of  Power  Plants 8vo,  5  oo 

Heat  and  Heat-engines 8vo,  5  oo 

Kent's  Steam  boiler  Economy 8vo,  4  oo 

Kneass's  Practice  and  Theory  of  the  Injector 8vo,  i  50 

MacCord's  Slide-valves 8vo,  2  oo 

Meyer's  Modern  Locomotive  Construction 4to,  10  oo 

Peabody's  Manual  of  the  Steam-engine  Indicator i2mo.  i  50 

Tables  of  the  Properties  of  Saturated  Steam  and  Other  Vapors 8vo,  i  oo 

Thermodynamics  of  the  Steam-engine  and  Other  Heat-engines 8vo,  5  oo 

Valve-gears  for  Steam-engines 8vo,  2  50 

Peabody  and  Miller's  Steam-boilers 8vo,  4  oo 

Pray's  Twenty  Years  with  the  Indicator Large  Svo,  2  50 

Pupin's  Thermodynamics  of  Reversible  Cycles  in  Gases  and  Saturated  Vapors. 

(Osterberg.) i2mo,  i   25 

Reagan's  Locomotives:   Simple   Compound,  and  Electric i2mo,  2  50 

Rontgen's  Principles  of  Thermodynamics.     (Du  Bois.) 8vo,  5  oo 

Sinclair's  Locomotive  Engine  Running  and  Management i2mo,  2  oo 

Smart's  Handbook  of  Engineering  Laboratory  Practice 12010,  2  50 

Snow's  Steam-boiler  Practice.  . 8vo,  3  oo 

Spangier's  Valve-gears 8vo,  2  50 

Notes  on  Thermodynamics i2mo,  i  oo 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering 8vo,  3  oo 

Thurston's  Handy  Tables 8vo.  i  50 

Manual  of  the  Steam-engine 2  vols.,  8vo,  10  oo 

Part  I.     History,  Structure,  and  Theory 8vo,  6  oo 

Part  II.     Design,  Construction,  and  Operation 8vo,  6  oo 

Handbook  of  Engine  and  Boiler  Trials,  and  the  Use  of  the  Indicator  and 

the  Prony  Brake 8vo,  5  oo 

Stationary  Steam-engines Svo,  2  50 

Steam-boiler  Explosions  in  Theory  and  in  Practice • i2mo,  i  50 

Manual  of  Steam-boilers,  their  Designs,  Construction,  and  Operation Svo,  5  oo 

Weisbach's  Heat,  Steam,  and  Steam-engines.     (Du  Bois.) Svo,  5  oo 

Whitham's  Steam-engine  Design « Svo,  5  oo 

Wilson's  Treatise  on  Steam-boilers.     (Flather.) i6mo,  2  50 

Wood's  Thermodynamics,  Heat  Motors,  and  Refrigerating  Machines.  .  .Svo,  4  oo 


MECHANICS  AND  MACHINERY. 

Barr's  Kinematics  of  Machinery Svo,  2  50 

Bovey's  Strength  of  Materials  and  Theory  of  Structures Svo,  7  50 

Chase's  The  Art  of  Pattern-making i2mo,  2  50 

Church's  Mechanics  of  Engineering Svo,  6  oo 

13 


Church's  Notes  and  Examples  in  Mechanics 8vo,  2  oo 

Compton's  First  Lessons  in  Metal-working i2mo,  i  50 

Compton  and  De  Groodt's  The  Speed  Lathe i2mo,  i  50 

Cromwell's  Treatise  on  Toothed  Gearing i2mo,  i  50 

Treatise  on  Belts  and  Pulleys i2mo,  i  50- 

Dana's  Text-book  of  Elementary  Mechanics  for  Colleges  and  Schools.  .  i2mo,  i  50 

Dingey's  Machinery  Pattern  Making i2mo,  2  oo 

Dredge's  Record  of  the   Transportation  Exhibits   Building  of   the   World's 

Columbian  Exposition  of  1893 4to  half  morocco,  5  oo 

Du  Bois's  Elementary  Principles  of  Mechanics : 

Vol.      I.     Kinematics 8vo,  3  50 

Vol.    II.     Statics 8vo,  4  oo 

Vol.  III.     Kinetics 8vo,  3  50 

Mechanics  of  Engineering.     Vol.    I Small  4to,  7  50 

Vol.  II Small  4to,  10  oo 

Durley's  Kinematics  of  Machines 8vo,  4  oo 

Fitzgerald's  Boston  Machinist i6mo,  i  oo 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  oo 

Rope  Driving i2mo,  2  oo 

Goss's  Locomotive  Sparks 8vo,  2  oo 

Hall's  Car  Lubrication i2mo,  i  oo 

Holly's  Art  of  Saw  Filing i8mo,  75 

James's  Kinematics  of  a  Point  and  the  Rational  Mechanics  of  a  Particle.     (In  press.) 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,  3  oo 

Johnson's  (L.  J.)  Statics  by  Graphic  and  Algebraic  Methods 8vo,  2  oo 

Jones's  Machine  Design:  • 

Part    I.     Kinematics  of  Machinery 8vo,  i  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

Kerr's  Power  and  Power  Transmission 8vo,  2  oo 

Lanza's  Applied  Mechanics 8vo,  7  50 

Leonard's  Machine  Shop,  Tools,  and  Methods.     (In  press.) 

Lorenz's  Modern  Refrigerating  Machinery.      (Pope,  Haven,  and  Dean.)      (In  press.) 

MacCord's  Kinematics;   or,  Practical  Mechanism 8vo,  5  oo 

Velocity  Diagrams 8vo,  i  50 

Maurer's  Technical  Mechanics 8vo,  4  oo 

Merriman's  Text-book  on  the  Mechanics  of  Materials 8vo,  4  oo 

*  Elements  of  Mechanics " i2mo,  i  oo 

*  Michie's  Elements  of  Analytical  Mechanics 8vo,  4  oo 

Reagan's  Locomotives:   Simple,  Compound,  and  Electric i2mo/  2  50 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  oo 

Richards's  Compressed  Air i2mo,  i   50 

Robinson's  Principles  of  Mechanism 8vo,  3  co 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     Vol.  1 8vo,  2  50 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  oo 

Sinclair's  Locomotive-engine  Running  and  Management 12 mo,  2  oo 

Smith's  (O.)  Press-working  of  Metals 8vo,  3  oo 

Smith's  (A.  W.)  Materials  of  Machines i2mo,  i  oo 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering 8vo,  3  oo 

Thurston's  Treatise  on  Friction  and   Lost  Y/ork  in    Machinery  and    Mill 

Work 8vo,  3  oo 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Lawc  of  Energetics.. 

i2mo,  i  oo 

Warren's  Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Weisbach's  Kinematics  and  Power  of  Transmission.    (Herrmann — Klein.  ).8vo,  5  oo 

Machinery  of  Transmission  and  Governors.      (Herrmann — Klein. ).8vo,  5  oo 

Wood's  Elements  of  Analytical  Mechanics 8vo,  3  oo 

Principles  of  Elementary  Mechanics i2mo,  i  25 

Turbines 8vo .  2  50 

The  World's  Columbian  Exposition  of  1893 4to,  i  oo 

14 


METALLURGY. 

Egleston's  Metallurgy  of  Silver,  Gold,  and  Mercury: 

Vol.    I.     Silver 8vo,  7  50 

Vol.  II.     Gold  and  Mercury 8vo,  7  50 

**  Iles's  Lead-smelting.     (Postage  9  cents  additional.) i2mo,  2  50 

Keep's  Cast  Iron 8vo,  z  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe 8vo,  i  50 

Le  Chatelier's  High-temperature  Measurements.  (Boudouard — Burgess. )i2mo,  3  oo 

Metcalf's  Steel.     A  Manual  for  Steel-users     i2mo,  2  oo. 

Smith's  Materials  of  Machines i2mo,  i  oo 

Thurston's  Materials  of  Engineering.     In  Three  Parts 8vo,  8  oa 

Part    II.     Iron  and  Steel 8vo,  3  50 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents. 8vo,  2  50 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  3  oa 

MINERALOGY. 

Barringer's  Description  of  Minerals  of  Commercial  Value.    Oblong,  morocco,  2  50 

Boyd's  Resources  of  Southwest  Virginia 8vo,  3  oo 

Map  of  Southwest  Virignia Pocket-book  form.  2  oo 

Brush's  Manual  of  Determinative  Mineralogy.     (Penfield.) 8vo,  4  oo 

Chester's  Catalogue  of  Minerals 8vo,  paper,  i  oo 

Cloth,  i  25 

Dictionary  of  the  Names  of  Minerals 8vo,  3  50 

Dana's  System  of  Mineralogy.  . Large  8vo,  half  leather,  12  50 

First  Appendix  to  Dana's  New  "  System  of  Mineralogy." Large  8vo,  i  oo 

Text-book  of  Mineralogy 8vo,  4  oo 

Minerals  and  How  to  Study  Them I2mo,  i  50 

Catalogue  of  American  Localities  of  Minerals Large  8vo,  i  oo 

Manual  of  Mineralogy  and  Petrography i2mo  2  oo 

Douglas's  Untechnical  Addresses  on  Technical  Subjects. i2mo,  i  oo 

Eakle's  Mineral  Tables 8vo,  i  25 

Egleston's  Catalogue  of  Minerals  and  Synonyms 8vo,  2  50 

Hussak's  The  Determination  of  Rock-forming  Minerals.    ( Smith.). Small  8vo,  2  oo 

Merrill's  Non-metallic  Minerals:   Their  Occurrence  and  Uses 8vo,  4  oo 

*  Penfieid's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

8vo.  paper,  o  50 
Rosenbusch's    Microscopical   Physiography   ot   the   Rock-makimg  Minerals 

(Iddings.) 8vo.  5  oo 

*  Tillman  s  Text-book  of  Important  Minerals  and  Rocks ...    .8vo.  2  oo 

Williams's  Manual  of  Lithology 8vo,  3  oo 

MINING. 

Beard's  Ventilation  of  Mines I2mo.  2  50 

Boyd's  Resources  of  Southwest  Virginia 8vo.  3  oo 

Map  of  Southwest  Virginia Pocket  book  form.  2  oo 

Douglas's  Untechnical  Addresses  on  Technical  Subjects i2mo.  i  oo 

*  Drinker's  Tunneling,  Explosive  Compounds,  and  Rock  Drills.  .4to,hf.mor  25  oo 

Eissler's  Modern  High  Explosives 8vo  4  oo 

Fowler's  Sewage  Works  Analyses i2tno  2  oo 

Goodyear's  Coal-mines  of  the  Western  Coast  of  the  United  States       .    .  i2mo.  2  50 

Ihlseng's  Manual  of  Mining 8vo.  5  oo 

**  Iles's  Lead-smelting.     (Postage  QC.  additional.) i2mo.  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe .8vo,  i  50 

O'DriscoU's  Notes  on  the  Treatment  of  Gold  Ores 8vo,  2  oo 

*  Walke's  Lectures  on  Explosives .  8vo,  4  oo 

Wilson's  Cyanide  Processes i2mo,  i  50 

Chlorination  Process i2mo,  i  50 

15 


Wilson's  Hydraulic  and  Placer  Mining '.....  i2mo,  2  oo 

Treatise  on  Practical  and  Theoretical  Mine  Ventilation T2mo.  i  25 

SANITARY  SCIENCE. 

FolwelFs  Sewerage.     (Designing,  Construction,  and  Maintenance.) 8vo,  3  oo 

Water-supply  Engineering 8vo,  4  oo 

Fuertes's  Water  and  Public  Health I2mo,  i  50 

Water-filtration  Works i2mo,  2  50 

Gerhard's  Guide  to  Sanitary  House-inspection i6mo,  i  oo 

Goodrich's  Economic  Disposal  of  Town's  Refuse DemySvo,  3  50 

Hazen's  Filtration  of  Public  Water-supplies 8vo,  3  oo 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control 8vo,  7  50 

Mason's  Water-supply.  (Considered  principally  from  a  Sanitary  Standpoint)  8vo,  4  oo 

Examination  of  Water.     (Chemical  and  Bacteriological.) I2mo,  i  25 

Merriman's  Elements  of  Sanitary  Engineering 8vo,  2  oo 

Ogden's  Sewer  Design i2mo,  2  oo 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis I2mo,  i  25 

*  Price's  Handbook  on  Sanitation I2mo,  i  50 

Richards's  Cost  of  Food.     A  Study  in  Dietaries I2mo,  i  oo 

Cost  of  Living  as  Modified  by  Sanitary  Science i2mo,  i  oo 

Richards  and  Woodman's  Air,  Water,  and   Food  from  a  Sanitary  Stand- 
point  8vo,  2  oo 

*  Richards  and  Williams's  The  Dietary  Computer 8vo,  i  50 

Rideal's  Sewage  and  Bacterial  Purification  of  Sewage 8vo,  3  50 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  oo 

Von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) i2mo,  i  oo 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Woodhull's  Notes  on  Military  Hygiene i6mo,  i  50 

MISCELLANEOUS. 

De  Fursac's  Manual  of  Psychiatry.  (Rosanoff  and  Collins.).  .  .  .Large  i2mo,  2  50 
Emmons's  Geological  Guide-book  of  the  Rocky  Mountain  Excursion  of  the 

International  Congress  of  Geologists Large  8vo,  i  50 

Ferrel's  Popular  Treatise  on  the  Winds 8vo.  4  oo 

Haines's  American  Railway  Management i2mo,  2  50 

ITott's  Composition,  Digestibility,  and  Nutritive  Value  of  Food.  Mounted  chart,  i  25 

Fallacy  of  the  Present  Theory  of  Sound i6mo,  i  oo 

Ricketts's  History  of  Rensselaer  Polytechnic  Institute,  1824-1894.  .Small  8vo,  3  oo 

Rostoski's  Serum  Diagnosis.  (Bolduan.). i2mo,  i  oo 

Rotherham's  Emphasized  New  Testament Large  8vo,  2  oo 

Steel's  Treatise  on  the  Diseases  of  the  Dog 8vo,  3  5° 

Totten's  Important  Question  in  Metrology 8vo,  2  50 

The  World's  Columbian  Exposition  of  1893 4to,  i  oo 

Von  Behring's  Suppression  of  Tuberculosis.  (Bolduan.) i2mo,  i  oo 

Winslow's  Elements  of  Applied  Microscopy i2mo,  i  50 

Worcester  and  Atkinson.  Small  Hospitals,  Establishment  and  Maintenance; 

Suggestions  for  Hospital  Architecture :  Plans  for  Small  Hospital .  i2mo,  i  25 

HEBREW  AND   CHALDEE  TEXT-BOOKS. 

Green's  Elementary  Hebrew  Grammar i2mo,  i  25 

Hebrew  Chrestomathy 8vo,  2  oo 

Gesenius's  Hebrew  and  Chaldee  Lexicon  to^tb«*OS3^8^ment  Scriptures. 

(Tregelles.) ^&**~Ptfjp*liL  4*0. half  morocco,  5  oo 

Lettems's  Hebrew  Bible ,/..:..  .P.F.  7"*  .—**  •  ^1 8v°'  2  *5 


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